Vector Algebra

Let \(\vec{a}=3 \hat{i}-\hat{j}+2 \hat{k}, \vec{b}=\vec{a} \times(\hat{i}-2 \hat{k})\) and \(\vec{c}=\vec{b} \times \hat{k}\). Then the projection of \(\vec{c}-2 \hat{j}\) on \(\vec{a}\) is:

[JEE Main 2025, 24 Jan (Shift 2)]

Let \(\vec{a}=\hat{i}+2\hat{j}+\hat{k}\) and \(\vec{b}=2\hat{i}+\hat{j}-\hat{k}\). Let \(\hat{c}\) be a unit vector in the plane of the vectors \(\vec{a}\) and \(\vec{b}\) and be perpendicular to \(\vec{a}\). Then such a vector \(\hat{c}\) is :

[JEE Main 2025, 8 Apr (Shift 1)]

\(\vec{a}=2\hat{i}-\hat{j}+3\hat{k},\overset{⇀}{b}=3\hat{i}-5\hat{j}+\hat{k}If\vec{a}\times \vec{c}=\vec{c}\times \vec{b}\\ and(\vec{a}+\vec{c}).(\vec{c}+\vec{b})=168.then|\vec{c}{|}^{2}=________\)

Let \(\vec{a}=\sqrt{7 } \hat{\imath}+\hat{\jmath}-\hat{k}\) and \(\vec{b}=\hat{j}+2 \hat{k}\). If \(\vec{r}\) is a vector such that \(\vec{r} \times \vec{a}+\vec{a} \times \vec{b}=\overrightarrow{0}\) and \(\vec{r} \cdot \vec{a}=0\), then \(|3 \vec{r}|^2\) is equal to:

[JEE Main 2026, 5 Apr (Shift 1)]

Vectors with position vector \(A= \quad 2 \hat{i}+3 n \hat{j}+2 \hat{k} \quad B=\) \(2 \hat{i}-2 \hat{j}+4p \hat{k}\), such that they are perpendicular and equidistance from origin
Find \(3 n+4 p\)

\(\vec{a}=2\hat{i}-\hat{j}+3\hat{k},\overset{⇀}{b}=3\hat{i}-5\hat{j}+\hat{k}If\vec{a}\times \vec{c}=\vec{c}\times \vec{b}\\ and(\vec{a}+\vec{c}).(\vec{c}+\vec{b})=168.then|\vec{c}{|}^{2}=________\)

Let \(A B C D\) be a tetrahedron such that the edges \(A B, A C\) and \(A D\) are mutually perpendicular. Let the areas of the triangles \(A B C, A C D\) and \(A D B\) be 5,6 and 7 square units respectively. Then the area (in square units) of the \(\triangle \mathrm{BCD}\) is equal to :

[JEE Main 2025, 2 Apr (Shift 1)]

Let \(\vec{a},\text{  }\vec{b}\) be two vectors, and let \(P, Q\) and \(R\) be the points with position vectors \(\vec{a},\text{  }\vec{b}\) and \(\vec{a}+\vec{b}\), respectively, with respect to the origin \(O\). If \(\left|\vec{a}+\vec{b}\right|=\sqrt{21},\text{ }\left|\vec{a}−\vec{b}\right|=3\), and \(\vec{a}\) and (\(\vec{a}−\vec{b}\)) are perpendicular to each other, then the area of the triangle \(OPR\) is

[JEE Advanced 2026]

Let \(\vec{a}=\hat{i}−2\hat{j}+3\hat{k},\vec{b}=2\hat{i}+\hat{j}−\hat{k},\vec{c}=\lambda \hat{i}+\hat{j}+\hat{k}\) and \(\vec{v}=\vec{a}\times \vec{b}\). If \(\vec{v}⋅\vec{c}=11\) and the length of the projection of \(\vec{b}\) on \(\vec{c}\) is \(p,\) then \(9{p}^{2}\) is equal to

[JEE Main 2026, 23 Jan (Shift 2)]

Let \(\vec{a}=-\hat{i}+\hat{j}+2\hat{k},\vec{b}=\hat{i}-\hat{j}-3\hat{k},\vec{c}=\vec{a}\times \vec{b}\) and \(\vec{d}=\vec{c} \times \vec{a}\). Then \((\vec{a}-\vec{b}) \cdot \vec{d}\) is equal to:

[JEE Main 2026, 23 Jan (Shift 1)]

Consider the vectors \(\vec{x}=\hat{\imath}+2 \hat{\jmath}+3 \hat{k}, \vec{y}=2 \hat{\imath}+3 \hat{\jmath}+\hat{k}\), and \(\vec{z}=3 \hat{\imath}+\hat{\jmath}+2 \hat{k}\) . For two distinct positive real numbers \(\alpha\) and \(\beta\), define\(\vec{X}=\alpha \vec{x}+\beta \vec{y}-\vec{z},\vec{Y}=\alpha \vec{y}+\beta \vec{z}-\vec{x},\text{ and }\vec{Z}=\alpha \vec{z}+\beta \vec{x}-\vec{y}\) If the vectors \(\vec{X}, \vec{Y}\), and \(\vec{Z}\) lie in a plane, then the value of \( \alpha+\beta-3 \) is_________.

[JEE Advanced 2025]

If \(\vec{OA}=\sqrt{3}\hat{ı}+\hat{ȷ},\vec{OB}=\hat{ı}+\sqrt{3}\hat{ȷ}\). If the distance of the point \(a\hat{ı}+(1-a)\hat{ȷ}\) from the angle bisector of \(\vec{OA}\) and \(\vec{OB}\) is \(\frac{9}{\sqrt{2}}\) the find \(a.\)

Let \(\vec{a}=\hat{i}+2\hat{j}+3\hat{k},\vec{b}=3\hat{i}+\hat{j}-\hat{k}\) and \(\vec{\mathrm{c}}\) be three vectors such that \(\vec{\mathrm{c}}\) is coplanar with \(\vec{\mathrm{a}}\text{ and }\vec{\mathrm{b}}.\) If the vector \(\vec{\mathrm{c}}\) is perpendicular to \(\vec{b}\) and \(\vec{\mathrm{a}}\cdot \vec{\mathrm{c}}=5,\) then \(|\vec{\mathrm{c}}|\) is equal to

[JEE Main 2025, 24 Jan (Shift 1)]

Let \(\vec{b}=\lambda \hat{i}+4 \hat{k}, \lambda>0\) and the projection vector of \(\vec{b}\) on \(\vec{a}=2 \hat{i}+2 \hat{j}-\hat{k}\) is \(\vec{c}\). If \(|\vec{a}+\vec{c}|=7\), then the area of the parallelogram formed by vector \(\vec{b}\) and \(\vec{c}\) is (in square units)

Let \(\vec{a}, \vec{b}, \vec{c}\) be three vectors such that \(\vec{a} \times \vec{b}=2(\vec{a} \times \vec{c})\). If \(|\vec{a}|=1,|\vec{b}|=4,|\vec{c}|=2\), and the angle between \(\vec{b}\) and \(\vec{c}\) is \(60^{\circ}\), then \(|\vec{a} \cdot \vec{c}|\) is equal to

[JEE Main 2026, 23 Jan (Shift 2)]

If \(\vec{OA}=\sqrt{3}\hat{ı}+\hat{ȷ},\vec{OB}=\hat{ı}+\sqrt{3}\hat{ȷ}\). If the distance of the point \(a\hat{ı}+(1-a)\hat{ȷ}\) from the angle bisector of \(\vec{OA}\) and \(\vec{OB}\) is \(\frac{9}{\sqrt{2}}\) the find \(a.\)

Let \(\vec{a}=4\hat{i}-\hat{j}+3\hat{k},\vec{b}=10\hat{i}+2\hat{j}-\hat{k}\) and a vector \(\vec{c}\) be such that \(2(\vec{a} \times \vec{b})+3(\vec{b} \times \vec{c})=\overrightarrow{0}\). If \(\vec{a} \cdot \vec{c}=15\), then \(\vec{c} \cdot(\hat{i}+\hat{j}-3 \hat{k})\) is equal to:

[JEE Main 2026, 8 Apr (Shift 2)]

Let the position vectors of three vertices of a triangle be \(4 \vec{p}+\vec{q}-3 \vec{r},-5 \vec{p}+\vec{q}+2 \vec{r}\) and \(2 \overrightarrow{\mathrm{p}}-\overrightarrow{\mathrm{q}}+2 \overrightarrow{\mathrm{r}}\). If the position vectors of the orthocenter and the circumcenter of the triangle are \(\frac{\vec{p}+\vec{q}+\vec{r}}{4}\) and \(\alpha \vec{p}+\beta \vec{q}+\gamma \vec{r}\) respectively, then \(\alpha+2 \beta+5 \gamma\) is equal to :

[JEE Main 2025, 24 Jan (Shift 2)]

Let \(\mathbf{a}=\hat{i}-\hat{k}, \mathbf{b}=x \hat{i}+\hat{j}+(1-x) \hat{k}\), and \(\mathbf{c}=y \hat{i}+x \hat{j}+(1+x-y) \hat{k}\). Then, \([\mathbf{a} \mathbf{b} \mathbf{~ c}]\) depends on:

If \(\vec{a}\) is nonzero vector such that its projections on the vectors \(2\hat{\mathrm{i}}-\hat{\mathrm{j}}+2\hat{\mathrm{k}},\hat{\mathrm{i}}+2\hat{\mathrm{j}}-2\hat{\mathrm{k}}\) and \(\hat{k}\)are equal, then a unit vector along \(\vec{a}\) is:

[JEE Main 2025, 2 Apr (Shift 1)]

Consider two vectors \(\vec{\mathrm{u}}=3\hat{\mathrm{i}}-\hat{\mathrm{j}}\) and \(\vec{\mathrm{v}}=2\hat{\mathrm{i}}+\hat{\mathrm{j}}-\lambda \hat{\mathrm{k}},\lambda >0\). The angle between them is given by \({\cos }^{-1}\left(\frac{\sqrt{5}}{2\sqrt{7}}\right)\). Let \(\vec{v}={\vec{v}}_{1}+{\vec{v}}_{2}\), where \({\vec{v}}_{1}\) is parallel to \(\vec{\mathrm{u}}\) and \({\vec{\mathrm{v}}}_{2}\) is perpendicular to \(\vec{\mathrm{u}}\). Then the value \({\left|{\vec{v}}_{1}\right|}^{2}+{\left|{\vec{v}}_{2}\right|}^{2}\) is equal to

[JEE Main 2025, 4 Apr (Shift 1)]

Let \(\vec{a}=\hat{i}+2\hat{j}+\hat{k}\) and \(\vec{b}=2\hat{i}+\hat{j}-\hat{k}\). Let \(\hat{c}\) be a unit vector in the plane of the vectors \(\vec{a}\) and \(\vec{b}\) and be perpendicular to \(\vec{a}\). Then such a vector \(\hat{c}\) is:

Let \(\vec{a}=2\hat{i}-\hat{j}+3\hat{k},\vec{b}=3\hat{i}-5\hat{j}+\hat{k}\text{ and }\vec{c}\)

be a vector such that \(\vec{\mathrm{a}}\times \vec{\mathrm{c}}=\vec{\mathrm{c}}\times \vec{\mathrm{b}}\) and

\(\left(\vec{a}+\vec{c}\right)\cdot \left(\vec{b}+\vec{c}\right)=168\). Then the maximum

value of \(|\vec{\mathrm{c}}{|}^{2}\) is:

[JEE Main 2025, 29 Jan (Shift 1)]

If \(\vec{a}\) is nonzero vector such that its projections on the vectors \(2\hat{\mathrm{i}}-\hat{\mathrm{j}}+2\hat{\mathrm{k}},\hat{\mathrm{i}}+2\hat{\mathrm{j}}-2\hat{\mathrm{k}}\) and \(\hat{k}\)are equal, then a unit vector along \(\vec{a}\) is:

[JEE Main 2025, 2 Apr (Shift 1)]

Let \(\vec{a}=3 \hat{\imath}+2 \hat{\jmath}-\hat{k}, \vec{b}=\vec{a} \times(\hat{\imath}-2 \hat{\jmath})\) and \(\vec{c}=\vec{b} \times \hat{k}\), then magnitude of projection of \(\vec{c}-2 \hat{\jmath}\) on \(\vec{a}\) is equal to

Let \(\mathbf{a}=\hat{i}-\hat{k}, \mathbf{b}=x \hat{i}+\hat{j}+(1-x) \hat{k}\), and \(\mathbf{c}=y \hat{i}+x \hat{j}+(1+x-y) \hat{k}\). Then, \([\mathbf{a} \mathbf{b} \mathbf{~ c}]\) depends on:

Let \(\vec{a}\) and \(\vec{b}\) be two unit vectors such that the angle between them is \(\frac{\pi}{3}\). If \(\lambda \vec{a}+2 \vec{b}\) and \(3 \vec{a}-\lambda \vec{b}\) are perpendicular to each other, then the number of values of \(\lambda\) in \([-1,3]\) is :

[JEE Main 2025, 22 Jan (Shift 2)]

Let \(\vec{\mathrm{a}}=2\hat{\mathrm{i}}-3\hat{\mathrm{j}}+\hat{\mathrm{k}},\vec{\mathrm{b}}=3\hat{\mathrm{i}}+2\hat{\mathrm{j}}+5\hat{\mathrm{k}}\) and a vector \(\vec{c}\) be such that \((\vec{\mathrm{a}}-\vec{\mathrm{c}})\times \vec{\mathrm{b}}=-18\hat{\mathrm{i}}-3\hat{\mathrm{j}}+12\hat{\mathrm{k}}\) and \(\vec{\mathrm{a}}\cdot \vec{\mathrm{c}}=3\). If \(\vec{\mathrm{b}}\times \vec{\mathrm{c}}=\vec{\mathrm{d}}\), then \(|\vec{\mathrm{a}}\cdot \vec{\mathrm{d}}|\) is equal to :

[JEE Main 2025, 2 Apr (Shift 2)]

For three unit vectors \(\vec{a}, \vec{b}, \vec{c}\) satisfying \(|\vec{a}-\vec{b}|^2+|\vec{b}-\vec{c}|^2+|\vec{c}-\vec{a}|^2=9\) and \(|2 \vec{a}+k \vec{b}+k \vec{c}|=3\), the positive value of \(k\) is

[JEE Main 2026, 28 Jan (Shift 1)]

If \(\vec{a}=\hat{\imath}+2 \hat{\jmath}+3 \hat{k}, \vec{b}=3 \hat{\imath}+\hat{\jmath}-\hat{k}\) and \(\vec{c}\) is coplanar with \(\vec{a}\) and \(\vec{b}\). Also \(\vec{a} \cdot \vec{c}=5\) and \(\vec{c}\) is perpendicular to \(\vec{b}\). Then \(|\vec{c}|\) is (24 Jan, Shift I, Memory Based)

Let \(\vec{a}\) and \(\vec{b}\) be two unit vectors such that the angle between them is \(\frac{\pi}{3}\). If \(\lambda \vec{a}+2 \vec{b}\) and \(3 \vec{a}-\lambda \vec{b}\) are perpendicular to each other, then the number of values of \(\lambda\) in \([-1,3]\) is :

[JEE Main 2025, 22 Jan (Shift 2)]

Let the position vectors of three vertices of a triangle be \(4 \vec{p}+\vec{q}-3 \vec{r},-5 \vec{p}+\vec{q}+2 \vec{r}\) and \(2 \overrightarrow{\mathrm{p}}-\overrightarrow{\mathrm{q}}+2 \overrightarrow{\mathrm{r}}\). If the position vectors of the orthocenter and the circumcenter of the triangle are \(\frac{\vec{p}+\vec{q}+\vec{r}}{4}\) and \(\alpha \vec{p}+\beta \vec{q}+\gamma \vec{r}\) respectively, then \(\alpha+2 \beta+5 \gamma\) is equal to :

[JEE Main 2025, 24 Jan (Shift 2)]

Let \(\vec{a}\) and \(\vec{b}\) be two unit vectors such that the angle between them is \(\frac{\pi}{3}\). If \(\lambda \vec{a}+2 \vec{b}\) and \(3 \vec{a}-\lambda \vec{b}\) are perpendicular to each other, then the number of values of \(\lambda\) in \([-1,3]\) is :

[JEE Main 2025, 22 Jan (Shift 2)]

Let \(\vec{a}\) and \(\vec{b}\) be the vectors of the same magnitude such that \(\frac{|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|}{|\vec{a}+\vec{b}|-|\vec{a}-\vec{b}|}=\sqrt{2}+1\). Then \(\frac{|\vec{a}+\vec{b}{|}^{2}}{|\vec{a}{|}^{2}}\) is :

[JEE Main 2025, 7 Apr (Shift 2)]

\(\vec{a}=2\hat{i}-\hat{j}+3\hat{k},\overset{⇀}{b}=3\hat{i}-5\hat{j}+\hat{k}If\vec{a}\times \vec{c}=\vec{c}\times \vec{b}\\ and(\vec{a}+\vec{c}).(\vec{c}+\vec{b})=168.then|\vec{c}{|}^{2}=________\)

Let \(\vec{b}=\lambda \hat{i}+4 \hat{k}, \lambda>0\) and the projection vector of \(\vec{b}\) on \(\vec{a}=2 \hat{i}+2 \hat{j}-\hat{k}\) is \(\vec{c}\). If \(|\vec{a}+\vec{c}|=7\), then the area of the parallelogram formed by vector \(\vec{b}\) and \(\vec{c}\) is (in square units)

Let the position vectors of three vertices of a triangle be \(4 \vec{p}+\vec{q}-3 \vec{r},-5 \vec{p}+\vec{q}+2 \vec{r}\) and \(2 \overrightarrow{\mathrm{p}}-\overrightarrow{\mathrm{q}}+2 \overrightarrow{\mathrm{r}}\). If the position vectors of the orthocenter and the circumcenter of the triangle are \(\frac{\vec{p}+\vec{q}+\vec{r}}{4}\) and \(\alpha \vec{p}+\beta \vec{q}+\gamma \vec{r}\) respectively, then \(\alpha+2 \beta+5 \gamma\) is equal to :

[JEE Main 2025, 24 Jan (Shift 2)]

Let \(\vec{a}=\hat{i}+2\hat{j}+3\hat{k},\vec{b}=3\hat{i}+\hat{j}-\hat{k}\) and \(\vec{\mathrm{c}}\) be three vectors such that \(\vec{\mathrm{c}}\) is coplanar with \(\vec{\mathrm{a}}\text{ and }\vec{\mathrm{b}}.\) If the vector \(\vec{\mathrm{c}}\) is perpendicular to \(\vec{b}\) and \(\vec{\mathrm{a}}\cdot \vec{\mathrm{c}}=5,\) then \(|\vec{\mathrm{c}}|\) is equal to

[JEE Main 2025, 24 Jan (Shift 1)]

Let \(\vec{a}\) and \(\vec{b}\) be the vectors of the same magnitude such that \(\frac{|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|}{|\vec{a}+\vec{b}|-|\vec{a}-\vec{b}|}=\sqrt{2}+1\). Then \(\frac{|\vec{a}+\vec{b}{|}^{2}}{|\vec{a}{|}^{2}}\) is :

[JEE Main 2025, 7 Apr (Shift 2)]

Let \(\vec{a}=2\hat{i}−5\hat{j}+5\hat{k}\) and \(\vec{b}=\hat{i}−\hat{j}+3\hat{k}\). If \(\vec{c}\) is a vector such that \(2(\vec{a} \times \vec{c})+3(\vec{b} \times \vec{c})=\overrightarrow{0}\) and \((\vec{a}-\vec{b}) \cdot \vec{c}=-97\), then \(|\vec{c} \times \hat{k}|^2\) is equal to

[JEE Main 2026, 24 Jan (Shift 2)]

Let \(\vec{a}=2\hat{i}+3\hat{j}+3\hat{k}\) and \(\vec{b}=6\hat{i}+3\hat{j}+3\hat{k}\). Then the square of the area of the triangle with adjacent sides determined by the vectors \((2 \vec{a}+3 \vec{b})\) and \((\vec{a}-\vec{b})\) is:

[JEE Main 2026, 6 Apr (Shift 2)]

Let \(\hat{a}\) be a unit vector perpendicular to the vectors \(\vec{b}=\hat{i}-2 \hat{j}+3 \hat{k}\) and \(\vec{c}=2 \hat{i}+3 \hat{j}-\hat{k}\) , and makes an angle of \(\cos ^{-1}\left(-\frac{1}{3}\right)\) with the vector \(\hat{i}+\hat{j}+\hat{k}\) . If \(\hat{a}\) makes an angle of \(\frac{\pi}{3}\) with the vector \(\hat{i}+\alpha \hat{j}+\hat{k}\) , then the value of \(\alpha\) is :

Let the position vectors of three vertices of a triangle be \(4 \vec{p}+\vec{q}-3 \vec{r},-5 \vec{p}+\vec{q}+2 \vec{r}\) and \(2 \overrightarrow{\mathrm{p}}-\overrightarrow{\mathrm{q}}+2 \overrightarrow{\mathrm{r}}\). If the position vectors of the orthocenter and the circumcenter of the triangle are \(\frac{\vec{p}+\vec{q}+\vec{r}}{4}\) and \(\alpha \vec{p}+\beta \vec{q}+\gamma \vec{r}\) respectively, then \(\alpha+2 \beta+5 \gamma\) is equal to :

[JEE Main 2025, 24 Jan (Shift 2)]

Consider a \(\triangle ABC\) where \(A (1,3,2), B (-2,8,0)\) and \(C (3,6,7)\). If the angle bisector of \(\angle BAC\) meets the line \(B C\) at \(D\), then the length of the projection of the vector \(\overrightarrow{A D}\) on the vector \(\overrightarrow{A C}\) is :EndFragment

[JEE Main 2024, 01 Feb (Shift 2)]

The set of all \(\alpha\), for which the vectors \(\vec{a}=\alpha t \hat{i}+6 \hat{j}-3 \hat{k}\) and \(\vec{b}=t \hat{i}-2 \hat{j}-2 \alpha t \hat{k}\) are inclined at an obtuse angle for all \(t \in R\), is

[JEE Main 2024, 8 Apr (Shift 1)]

The set of all \(\alpha\), for which the vectors \(\vec{a}=\alpha t \hat{i}+6 \hat{j}-3 \hat{k}\) and \(\vec{b}=t \hat{i}-2 \hat{j}-2 \alpha t \hat{k}\) are inclined at an obtuse angle for all \(t \in R\), is

[JEE Main 2024, 8 Apr (Shift 1)]

Let the vectors \(\vec{a}=-\hat{i}+\hat{j}+3 \hat{k}\) and \(\vec{b}=\hat{i}+3 \hat{j}+\hat{k}\). For some \(\lambda ,\mu \in R\), let \(\vec{c}=\lambda \vec{a}+\mu \vec{b}\). If \(\vec{c} \cdot(3 \hat{i}-6 \hat{j}+2 \hat{k})=10\) and \(\vec{c} \cdot(\hat{i}+\hat{j}+\hat{k})=-2\), then \(|\vec{c}|^2\) is equal to:

[JEE Main 2026, 2 Apr (Shift 2)]

Let \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) be three points in \(x y\)-plane, whose position vector are given by\(\sqrt{3} \hat{i}+\hat{j}, \hat{i}+\sqrt{3} \hat{j}\) and \(a \hat{i}+(1-a) \hat{j}\) respectively with respect to the origin \(O.\) If the distance of the point \(C\) from the line bisecting the angle between the vectors \(\overrightarrow{\mathrm{OA}}\) and \(\overrightarrow{\mathrm{OB}}\) is \(\frac{9}{\sqrt{2}}\), then the sum of all the possible values of \(a\) is :

[JEE Main 2025, 28 Jan (Shift 2)]

Let \(\vec{a}=3 \hat{\imath}+2 \hat{\jmath}-\hat{k}, \vec{b}=\vec{a} \times(\hat{\imath}-2 \hat{\jmath})\) and \(\vec{c}=\vec{b} \times \hat{k}\), then magnitude of projection of \(\vec{c}-2 \hat{\jmath}\) on \(\vec{a}\) is equal to

Vectors with position vector \(A= \quad 2 \hat{i}+3 n \hat{j}+2 \hat{k} \quad B=\) \(2 \hat{i}-2 \hat{j}+4p \hat{k}\), such that they are perpendicular and equidistance from origin
Find \(3 n+4 p\)

Let \(\hat{a}\) be a unit vector perpendicular to the vectors \(\vec{b}=\hat{i}-2 \hat{j}+3 \hat{k}\) and \(\vec{c}=2 \hat{i}+3 \hat{j}-\hat{k}\) , and makes an angle of\(\cos ^{-1}\left(-\frac{1}{3}\right)\) with the vector \(\hat{i}+\hat{j}+\hat{k}\) . If \(\hat{a}\) makes an angle of \(\frac{\pi}{3}\) with the vector \(\hat{i}+\alpha \hat{j}+\hat{k}\) , then the value of \(\alpha\) is :

[JEE Main 2025, 29 Jan (Shift 2)]

Let \(\vec{a}=3 \hat{i}-\hat{j}+2 \hat{k}, \vec{b}=\vec{a} \times(\hat{i}-2 \hat{k})\) and \(\vec{c}=\vec{b} \times \hat{k}\). Then the projection of \(\vec{c}-2 \hat{j}\) on \(\vec{a}\) is :

[JEE Main 2025, 24 Jan (Shift 2)]

Let \(A(x, y, z)\) be a point in \(xy-\)plane, which is equidistant from three points \((0, 3, 2), (2, 0, 3)\) and \((0, 0, 1).\)
Let \(B = (1, 4, –1)\) and \(C = (2, 0, –2).\) Then among the statements

\((S1) : \Delta ABC\) is an isosceles right angled triangle and
\((S2) :\) the area of \(∆\mathrm{ABC}\text{ is }\frac{9\sqrt{2}}{2}.\)

[JEE Main 2025, 28 Jan (Shift 1)]

Let \(\vec{a}=\hat{i}+\alpha \hat{j}+\beta \hat{k}, \alpha, \beta \in R\). Let a vector \(\vec{b}\) be such that the angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{4}\) and \(|\vec{b}|^2=6\). If \(\vec{a} \cdot \vec{b}=3 \sqrt{2}\), then the value of \(\left(\alpha^2+\beta^2\right)|\vec{a} \times \vec{b}|^2\) is equal to

[JEE Main 2024, 30 Jan (Shift 2)]

Let \(\vec{a}=\hat{i}+\alpha \hat{j}+\beta \hat{k}, \alpha, \beta \in R\). Let a vector \(\vec{b}\) be such that the angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{4}\) and \(|\vec{b}|^2=6\). If \(\vec{a} \cdot \vec{b}=3 \sqrt{2}\), then the value of \(\left(\alpha^2+\beta^2\right)|\vec{a} \times \vec{b}|^2\) is equal to

[JEE Main 2024, 30 Jan (Shift 2)]

Let the point A divide the line segment joining the points \(\mathrm{P}(-1,-1,2)\) and \(\mathrm{Q}(5,5,10)\) internally in the ratio \(\mathrm{r}: 1(\mathrm{r}>0)\). If O is the origin and \((\overrightarrow{\mathrm{OQ}} \cdot \overrightarrow{\mathrm{OA}})-\frac{1}{5}|\overrightarrow{\mathrm{OP}} \times \overrightarrow{\mathrm{OA}}|^2=10\), then the value of r is :

[JEE Main 2025, 23 Jan (Shift 2)]

Triangle ABC with vertices have position vectors \(2 \vec{p}-3 \vec{q}+2 \vec{r}, \vec{p}-\vec{q}+3 \vec{r},-\vec{p}+2 \vec{q}+5 \vec{r}\) and orthocentre's position vector is \(\frac{\vec{p}+\vec{q}+\vec{r}}{4}\) then find position vector of circumcenter.

Let ABCD be a tetrahedron such that the edges AB, AC and AD are mutually perpendicular. Let the areas of the triangles ABC, ACD and ADB be 5,6 and 7 square units respectively. Then the area (in square units) of the \(∆\mathrm{BCD}\) is equal to :

[JEE Main 2025, 2 Apr (Shift 1)]

Let \(O\) be the origin, \(\overrightarrow{O P}=\vec{a}\) and \(\overrightarrow{O Q}=\vec{b}\). If \(R\) is the point on \(\overrightarrow{O P}\) such that \(\overrightarrow{O P}=5 \overrightarrow{O R}\), and \(M\) is the point such that \(\overrightarrow{O Q}=5 \overrightarrow{R M}\), then \(\overrightarrow{P M}\) is equal to:

[JEE Main 2026, 5 Apr (Shift 2)]

If the components of \(\overrightarrow{\mathrm{a}}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}\) along and perpendicular to \(\overrightarrow{\mathrm{b}}=3 \hat{i}+\hat{j}-\hat{k}\) respectively, are \(\frac{16}{11}(3 \hat{i}+\hat{j}-\hat{k})\) and \(\frac{1}{11}(-4 \hat{i}-5 \hat{j}-17 \hat{k})\), then \(\alpha^2+\beta^2+\gamma^2\) is equal to :

[JEE Main 2025, 28 Jan (Shift 2)]

Let the angle \(\theta ,0<\theta <\frac{\pi }{2}\) between two unit vectors \(\hat{\mathrm{a}}\) and \(\hat{\mathrm{b}}\) be \({\sin }^{-1}\left(\frac{\sqrt{65}}{9}\right)\). If the vector \(\vec{\mathrm{c}}=3\hat{a}+6\hat{b}+9(\hat{a}\times \hat{b})\), then the value of \(9(\vec{\mathrm{c}}\cdot \hat{a})-3(\vec{\mathrm{c}}\cdot \hat{\mathrm{b}})\) is

[JEE Main 2025, 7 Apr (Shift 1)]

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three non-zero vectors such that \(\vec{b}\) and \(\vec{c}\) are non-collinear. If \(\vec{a}+5 \vec{b}\) is collinear with \(\vec{c}, \vec{b}+6 \vec{c}\) is collinear with \(\vec{a}\) and \(\vec{a}+\alpha \vec{b}+\beta \vec{c}=\overrightarrow{0}\),then \(\alpha+\beta\) is equal to

[JEE Main 2024, 29 Jan (Shift 1)]

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three non-zero vectors such that \(\vec{b}\) and \(\vec{c}\) are non-collinear. If \(\vec{a}+5 \vec{b}\) is collinear with \(\vec{c}, \vec{b}+6 \vec{c}\) is collinear with \(\vec{a}\) and \(\vec{a}+\alpha \vec{b}+\beta \vec{c}=\overrightarrow{0}\),then \(\alpha+\beta\) is equal to

[JEE Main 2024, 29 Jan (Shift 1)]

Let \(\vec{a}=3 \hat{i}-\hat{j}+2 \hat{k}, \vec{b}=\vec{a} \times(\hat{i}-2 \hat{k})\) and \(\vec{c}=\vec{b} \times \hat{k}\). Then the projection of \(\vec{c}-2 \hat{j}\) on \(\vec{a}\) is :

[JEE Main 2025, 24 Jan (Shift 2)]

Let a unit vector \(\hat{u}=x \hat{i}+y \hat{j}+z \hat{k}\) make angles \(\frac{\pi}{2}, \frac{\pi}{3}\) and \(\frac{2 \pi}{3}\) with the vectors \(\frac{1}{\sqrt{2}} \hat{i}+\frac{1}{\sqrt{2}} \hat{k}, \frac{1}{\sqrt{2}} \hat{j}+\frac{1}{\sqrt{2}} \hat{k}\) and \(\frac{1}{\sqrt{2}} \hat{i}+\frac{1}{\sqrt{2}} \hat{j}\) respectively. If \(\vec{v}=\frac{1}{\sqrt{2}} \hat{i}+\frac{1}{\sqrt{2}} \hat{j}+\frac{1}{\sqrt{2}} \hat{k}\), then \(|\hat{u}-\vec{v}|^2\) is equal to

[JEE Main 2024, 29 Jan (Shift 2)]

Let a unit vector \(\hat{u}=x \hat{i}+y \hat{j}+z \hat{k}\) make angles \(\frac{\pi}{2}, \frac{\pi}{3}\) and \(\frac{2 \pi}{3}\) with the vectors \(\frac{1}{\sqrt{2}} \hat{i}+\frac{1}{\sqrt{2}} \hat{k}, \frac{1}{\sqrt{2}} \hat{j}+\frac{1}{\sqrt{2}} \hat{k}\) and \(\frac{1}{\sqrt{2}} \hat{i}+\frac{1}{\sqrt{2}} \hat{j}\) respectively. If \(\vec{v}=\frac{1}{\sqrt{2}} \hat{i}+\frac{1}{\sqrt{2}} \hat{j}+\frac{1}{\sqrt{2}} \hat{k}\), then \(|\hat{u}-\vec{v}|^2\) is equal to

[JEE Main 2024, 29 Jan (Shift 2)]

Let \(\vec{\mathrm{a}}=2\hat{\mathrm{i}}-3\hat{\mathrm{j}}+\hat{\mathrm{k}},\vec{\mathrm{b}}=3\hat{\mathrm{i}}+2\hat{\mathrm{j}}+5\hat{\mathrm{k}}\) and a vector \(\vec{c}\) be such that \((\vec{\mathrm{a}}-\vec{\mathrm{c}})\times \vec{\mathrm{b}}=-18\hat{\mathrm{i}}-3\hat{\mathrm{j}}+12\hat{\mathrm{k}}\) and \(\vec{\mathrm{a}}\cdot \vec{\mathrm{c}}=3\). If \(\vec{\mathrm{b}}\times \vec{\mathrm{c}}=\vec{\mathrm{d}}\), then \(|\vec{\mathrm{a}}\cdot \vec{\mathrm{d}}|\) is equal to :

[JEE Main 2025, 2 Apr (Shift 2)]

Let \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) be three points in \(x y\)-plane, whose position vector are given by\(\sqrt{3} \hat{i}+\hat{j}, \hat{i}+\sqrt{3} \hat{j}\) and \(a \hat{i}+(1-a) \hat{j}\) respectively with respect to the origin O . If the distance of the point C from the line bisecting the angle between the vectors \(\overrightarrow{\mathrm{OA}}\) and \(\overrightarrow{\mathrm{OB}}\) is \(\frac{9}{\sqrt{2}}\), then the sum of all the possible values of \(a\) is :

[JEE Main 2025, 28 Jan (Shift 2)]

Let \(\vec{a}=3 \hat{i}-\hat{j}+2 \hat{k}, \vec{b}=\vec{a} \times(\hat{i}-2 \hat{k})\) and \(\vec{c}=\vec{b} \times \hat{k}\). Then the projection of \(\vec{c}-2 \hat{j}\) on \(\vec{a}\) is :

[JEE Main 2025, 24 Jan (Shift 2)]

Let \(\vec{a}\) and \(\vec{b}\) be two unit vectors such that the angle between them is \(\frac{\pi}{3}\). If \(\lambda \vec{a}+2 \vec{b}\) and \(3 \vec{a}-\lambda \vec{b}\) are perpendicular to each other, then the number of values of \(\lambda\) in \([-1,3]\) is :

[JEE Main 2025, 22 Jan (Shift 2)]

Let \(\vec{a}=2 \hat{i}-\hat{j}+3 \hat{k}, \quad \vec{b}=3 \hat{i}-5 \hat{j}+\widehat{k}\) and \(\vec{c} \) be a vector such that \(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{b}} \) and \((\vec{a}+\vec{c}) \cdot(\vec{b}+\vec{c})=168 \). Then the maximum value of \(|\overrightarrow{\mathrm{c}}|^2 \) is:

[JEE Main 2025, 29 Jan (Shift 1)]

Let \(\vec{a}=2\hat{i}−\hat{j}−\hat{k},\vec{b}=\hat{i}+3\hat{j}−\hat{k}\) and \(\vec{c}=2\hat{i}+\hat{j}+3\hat{k}\). Let \(\vec{v}\) be the vector in the plane of the vectors \(\vec{a}\) and \(\vec{b}\), such that the length of its projection on the vector \(\vec{c}\) is \(\frac{1}{\sqrt{14}}\). Then \(|\vec{v}|^2\) is equal to

[JEE Main 2026, 24 Jan (Shift 2)]

Let \(A(x, y, z)\) be a point in \(xy-\)plane, which is equidistant from three points \((0, 3, 2), (2, 0, 3)\) and \((0, 0, 1).\)
Let \(B = (1, 4, –1)\) and \(C = (2, 0, –2).\) Then among the statements

\((S_1) : \Delta ABC\) is an isosceles right angled triangle and
\((S_2) :\) the area of \(∆\mathrm{ABC}\text{ is }\frac{9\sqrt{2}}{2}.\)

[JEE Main 2025, 28 Jan (Shift 1)]

Let \(\hat{u}\) and \(\hat{v}\) be unit vectors inclined at an acute angle such that \(|\hat{u}\times \hat{v}|=\frac{\sqrt{3}}{2}\). If \(\vec{A}=\lambda \hat{u}+\hat{v}+\left(\hat{u}\times \hat{v}\right)\), then \(\lambda\) is equal to:

[JEE Main 2026, 4 Apr (Shift 2)]

If \(\vec{a}=\hat{\imath}+2 \hat{\jmath}+3 \hat{k}, \vec{b}=3 \hat{\imath}+\hat{\jmath}-\hat{k}\) and \(\vec{c}\) is coplanar with \(\vec{a}\) and \(\vec{b}\). Also \(\vec{a} \cdot \vec{c}=5\) and \(\vec{c}\) is perpendicular to \(\vec{b}\). Then \(|\vec{c}|\) is (24 Jan, Shift I, Memory Based)

Let the point A divide the line segment joining the points \(\mathrm{P}(-1,-1,2)\) and \(\mathrm{Q}(5,5,10)\) internally in the ratio \(\mathrm{r}: 1(\mathrm{r}>0)\). If O is the origin and \((\overrightarrow{\mathrm{OQ}} \cdot \overrightarrow{\mathrm{OA}})-\frac{1}{5}|\overrightarrow{\mathrm{OP}} \times \overrightarrow{\mathrm{OA}}|^2=10\), then the value of r is :

[JEE Main 2025, 23 Jan (Shift 2)]

Let \(\vec{a}=2\hat{i}+\hat{j}−2\hat{k},\vec{b}=\hat{i}+\hat{j}\) and \(\vec{c}=\vec{a}\times \vec{b}.\) Let \(\vec{d}\) be a vector such that \(|\vec{d}-\vec{a}|=\sqrt{11},|\vec{c} \times \vec{d}|=3\) and the angle between \(\vec{c}\) and \(\vec{d}\) is \(\frac{\pi }{4}.\) Then \(\vec{a}⋅\vec{d}\) is equal to

Triangle ABC with vertices have position vectors \(2 \vec{p}-3 \vec{q}+2 \vec{r}, \vec{p}-\vec{q}+3 \vec{r},-\vec{p}+2 \vec{q}+5 \vec{r}\) and orthocentre's position vector is \(\frac{\vec{p}+\vec{q}+\vec{r}}{4}\) then find position vector of circumcenter.

Consider two vectors \(\vec{\mathrm{u}}=3\hat{\mathrm{i}}-\hat{\mathrm{j}}\) and \(\vec{\mathrm{v}}=2\hat{\mathrm{i}}+\hat{\mathrm{j}}-\lambda \hat{\mathrm{k}},\lambda >0\). The angle between them is given by \({\cos }^{-1}\left(\frac{\sqrt{5}}{2\sqrt{7}}\right)\). Let \(\vec{v}={\vec{v}}_{1}+{\vec{v}}_{2}\), where \({\vec{v}}_{1}\) is parallel to \(\vec{\mathrm{u}}\) and \({\vec{\mathrm{v}}}_{2}\) is perpendicular to \(\vec{\mathrm{u}}\). Then the value \({\left|{\vec{v}}_{1}\right|}^{2}+{\left|{\vec{v}}_{2}\right|}^{2}\) is equal to

If \(\vec{a}\)and \(\vec{b}\) are two vectors such that \(|\vec{a}|=2\) and \(|\vec{b}|=3\), then the maximum value of \( 3|(3 \vec{a}+2 \vec{b})|+4|(3 \vec{a}-2 \vec{b})| \)is:

[JEE Main 2026, 2 Apr (Shift 1)]

If the components of \(\overrightarrow{\mathrm{a}}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}\) along and perpendicular to \(\overrightarrow{\mathrm{b}}=3 \hat{i}+\hat{j}-\hat{k}\) respectively, are \(\frac{16}{11}(3 \hat{i}+\hat{j}-\hat{k})\) and \(\frac{1}{11}(-4 \hat{i}-5 \hat{j}-17 \hat{k})\), then \(\alpha^2+\beta^2+\gamma^2\) is equal to :

[JEE Main 2025, 28 Jan (Shift 2)]

Let three vectors \(\vec{a}, \vec{b}\) and \(\vec{c}\) be such that \(\vec{a} \times \vec{b}=\vec{c}, \vec{b} \times \vec{c}=\vec{a}\) and \(|\vec{a}|=2\). Then which one of the following is not true?

[JEE Main 2021, 22 Jul (Shift 2)]

Let \( \vec{a} \) and \( \vec{b} \) be two vectors such that \( |2 \vec{a}+3 \vec{b}|=|3 \vec{a}+\vec{b}| \) and the angle between \( \vec{a} \) and \( \vec{b} \) is \( 60^{\circ} \). If \( \frac{1}{8} \vec{a} \) is a unit vector, then \( |\vec{b}| \) is equal to:

[JEE Main 2021, 31 Aug (Shift 1)]

Let \(\vec{a}=\hat{i}+\hat{j}+2 \hat{k}\) and \(\vec{b}=-\hat{i}+2 \hat{j}+3 \hat{k}\). Then the vector product \((\vec{a}+\vec{b}) \times((\vec{a} \times((\vec{a}-\vec{b}) \times \vec{b})) \times \vec{b})\) is equal to:

[JEE Main 2021, 27 Jul (Shift 1)]

If \(\vec{a}=\hat{i}+2 \hat{k}, \vec{b}=\hat{i}+\hat{j}+\hat{k}, \vec{c}=7 \hat{i}-3 \hat{j}+4 \hat{k}\), \(\vec{r} \times \vec{b}+\vec{b} \times \vec{c}=\overrightarrow{0}\) and \(\vec{r} \cdot \vec{a}=0\). Then \(\vec{r} \cdot \vec{c}\) is equal to

[JEE Main 2023, 29 Jan (Shift 2)]

Let a vector \( \vec{a} \) be coplanar with vectors \( \vec{b}=2 \hat{i}+\hat{j}+\hat{k} \) and \( \vec{c}=\hat{i}-\hat{j}+\hat{k} \). If \( \vec{a} \) is perpendicular to \( \vec{d}=3 \hat{i}+2 \hat{j}+6 \hat{k} \) and \( |\vec{a}|=\sqrt{10} \). Then a possible value of \( [\vec{a} \ \vec{b} \ \vec{c}] \)+ \( [\vec{a} \ \vec{b} \ \vec{d}] \)+ \( [\vec{a} \ \vec{c} \ \vec{d}] \) is equal to:

If the four points, whose position vectors are

\(3\hat{i}−4\hat{j}+2\hat{k},\hat{i}+2\hat{j}−\hat{k},−2\hat{i}−\hat{j}+3\hat{k}\) and \(5\hat{i}−2\alpha \hat{j}+4\hat{k}\) are coplanar, then \(\alpha\) is equal to

Let \(O\) be the origin and the position vector of the point \(P\) be \(-\hat{i}-2 \hat{j}+3 \hat{k}\). If the position vectors of the points \(A\), \(B\) and \(C\) are \(-2 \hat{i}+\hat{j}-3 k, 2 \hat{i}+4 \hat{j}-2 \hat{k}\) and \(-4 \hat{i}+2 \hat{j}-\hat{k}\) respectively, then the projection of the vector \(\overrightarrow{O P}\) on a vector perpendicular to the vectors \(\overrightarrow{A B}\) and \(\overrightarrow{A C}\) is

[JEE Main 2023, 10 Apr (Shift 1)]

If the four points, whose position vectors are \(3 \hat{i}-4 \hat{j}+2 \hat{k}, \hat{i}+2 \hat{j}-\hat{k},-2 \hat{i}-\hat{j}+3 \hat{k}\) and \(5 \hat{i}-2 \alpha \hat{j}+4 \hat{k}\) are coplanar, then \(\alpha\) is equal to]

Let \(\vec{a}=2\hat{i}+\hat{j}-2\hat{k}\) and \(\vec{b}=\hat{i}+\hat{j}\). If \(\vec{c}\) is a vector such that \(\vec{a}\cdot \vec{c}=|\vec{c}|,|\vec{c}-\vec{a}|=2\sqrt{2}\) and the angle between \((\vec{a}\times \vec{b})\) and \(\vec{c}\) is \({30}^{0}\), then the value of \(|(\vec{a}\times \vec{b})\times \vec{c}|\) is :

[JEE Main 2021, 20 Jul (Shift 1)]

Let the vectors \(\vec{u}_1=\hat{i}+\hat{j}+a \hat{k}, \vec{u}_2=\hat{i}+b \hat{j}+\hat{k}\) and \(\vec{u}_3=c \hat{i}+\hat{j}+\hat{k}\) be coplanar. If the vector \(\vec{v}_1=(a+b) \hat{i}+c \hat{j}+c \hat{k}, \vec{v}_2=a \hat{i}+(b+c) \hat{j}+a \hat{k}\) and \(\vec{v}_3=b \hat{i}+b \hat{j}+(c+a) \hat{k}\) are also coplanar, then \(6(a+b+c)\) is equal to

[JEE Main 2023, 8 Apr (Shift 2)]

If vectors \(\vec{a}_1=x \hat{i}-\hat{j}+\hat{k}\) and \(\vec{a}_2=\hat{i}+y \hat{j}+z \hat{k}\) are collinear, then a possible unit vector parallel to the vector \(x \hat{i}+y \hat{j}+z \hat{k}\) is:

[JEE Main 2021, 26 Feb (Shift 2)]

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three unit vectors such that \(\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}\). If \(\lambda=\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}\) and \(\vec{d}=\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}\), then the ordered pair, \((\lambda, \vec{d})\) is equal to:

[JEE Main 2020, 7 Jan (Shift 2)]

Let the position vectors of the points \(P, Q, R\) and \(S\) be \(\vec{a}=\hat{i}+2 \hat{j}-5 \hat{k}, \vec{b}=3 \hat{i}+6 \hat{j}+3 \hat{k}, \vec{c}=\frac{17}{5} \hat{i}+\frac{16}{5} \hat{j}+7 \hat{k}\) and \(\vec{d}=2 \hat{i}+\hat{j}+\hat{k}\), respectively. Then which of the following statements is true?

[JEE Advanced 2023]

Let \(\vec{a}=2 \hat{i}-7 \hat{j}+5 \hat{k}, \vec{b}=\hat{i}+\hat{k}\) and \(\vec{c}=\hat{i}+2 \hat{j}-3 \hat{k}\) be three given vectors. If \(\vec{r}\) is a vector such that \(\vec{r} \times \vec{a}=\vec{c} \times \vec{a}\) and \(\vec{r} \cdot \vec{b}=0\), then \(|\vec{r}|\) is equal to:

[JEE Main 2023, 1 Feb (Shift 2)]

Let \(\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{b}=\hat{i}-2 \hat{j}-2 \hat{k}\) and \(\vec{c}=-\hat{i}+4 \hat{j}+3 \hat{k}\). If \(\vec{d}\) is a vector perpendicular to both \(\vec{b}\) and \(\vec{c}\) and \(\vec{a} \cdot \vec{d}=18\). Then \(|\vec{a} \times \vec{d}|^2\) is equal to:

[JEE Main 2023, 6 Apr (Shift 1)]

Let \(|\vec{a}|=2,|\vec{b}|=3\) and the angle between the vectors \(\vec{a}\) and \(\vec{b}\) be \(\frac{\pi}{4}\). Then \(|(\vec{a}+2 \vec{b}) \times(2 \vec{a}-3 \vec{b})|^2\) is equal to

[JEE Main 2023, 13 Apr (Shift 2)]

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three non zero vectors such that \(\vec{b} \cdot \vec{c}=0\) and \(\vec{a} \times(\vec{b} \times \vec{c})=\frac{\vec{b}-\vec{c}}{2}\). If \(\vec{d}\) be a vector such that \(\vec{b} \cdot \vec{d}=\vec{a} \cdot \vec{b}\), then \((\vec{a} \times \vec{b}) \cdot(\vec{c} \times \vec{d})\) is equal to

[JEE Main 2023, 25 Jan (Shift 1)]

If \(\vec{a}\) and \(\vec{b}\) are perpendicular, then \(\vec{a} \times(\vec{a} \times(\vec{a} \times(\vec{a} \times \vec{b})))\) is equal to:

[JEE Main 2021, 26 Feb (Shift 1)]

Let for a triangle \(A B C\),
\(\overrightarrow{A B}=-2 \hat{i}+\hat{j}+3 \hat{k} \)

\(\overrightarrow{C B}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k} \)

\(\overrightarrow{C A}=4 \hat{i}+3 \hat{j}+\delta \hat{k}\)
If \(\delta>0\) and the area of the triangle \(A B C\) is \(5 \sqrt{6}\), then \(\overrightarrow{C B} \cdot \overrightarrow{C A}\) is equal to

[JEE Main 2023, 13 Apr (Shift 2)]

The area of the quadrilateral ABCD with vertices A(2, 1, 1), B(1, 2, 5), C(–2, –3, 5) and D(1, –6, –7) is equal to

[JEE Main 2023, 08 Apr (Shift 2)]

Let \(\vec{a}\) be a non-zero vector parallel to the line of intersection of the two planes described by \(\hat{i}+\hat{j}, \hat{i}+\hat{k}\) and \(\hat{i}-\hat{j}, \hat{j}-\hat{k}\). If \(\theta\) is the angle between the vector \(\vec{a}\) and the vector \(\vec{b}=2 \hat{i}-2 \hat{j}+\hat{k}\) and \(\vec{a} \cdot \vec{b}=6\) then the ordered pair \((\theta,|\vec{a} \times \vec{b}|)\) is equal to

[JEE Main 2023, 11 Apr (Shift 1)]

Let \(\vec{a}=2 \hat{i}-\hat{j}+2 \hat{k}\) and \(\vec{b}=\hat{i}+2 \hat{j}-\hat{k}\). Let a vector \(\vec{v}\) be in the plane containing \(\vec{a}\) and \(\vec{b}\). If \(\vec{v}\) is perpendicular to the vector \(3 \hat{i}+2 \hat{j}-\hat{k}\) and its projection on \(\vec{a}\) is 19 units, then \(|2 \vec{v}|^2\) is equal to__________

[JEE Main 2021, 1 Sep (Shift 2)]

If the four points, whose position vectors are \(3 \hat{i}-4 \hat{j}+2 \hat{k}, \hat{i}+2 \hat{j}-\hat{k}, -2 \hat{i}-\hat{j}+3 \hat{k}\) and \(5 \hat{i}-2 \alpha \hat{j}+4 \hat{k}\) are coplanar, then \(\alpha\) is equal to

[JEE Main 2023, 25 Jan (Shift 2)]

In a triangle \(A B C\), if \(|\overrightarrow{B C}|=3,|\overrightarrow{C A}|=5\) and \(|\overrightarrow{B A}|=7\), then the projection of the vector \(\overrightarrow{B A}\) on \(\overrightarrow{B C}\) is equal to:

[JEE Main 2021, 20 Jul (Shift 2)]

Let \(\vec{w}=\hat{ı}+\hat{ȷ}-2\hat{k}\), and \(\vec{u}\) and \(\vec{v}\) be two vectors, such that \(\vec{u}\times \vec{v}=\vec{w}\) and \(\vec{v}\times \vec{w}=\vec{u}\). Let \(\alpha ,\beta ,\gamma\), and \(t\) be real numbers such that \(\vec{u}=\alpha \hat{ı}+\beta \hat{ȷ}+\gamma \hat{k},-t\alpha +\beta +\gamma =0,\alpha -t\beta +\gamma =0,\)

\(\text{ and }\alpha +\beta -t\gamma =0\)

Match each entry in List-I to the correct entry in List-II and choose the correct option.

If the vectors \(\vec{a}=\lambda \hat{i}+\mu \hat{j}+4 \hat{k}, \vec{b}=-2 \hat{i}+4 \hat{j}-2 \hat{k}\) and \(\vec{c}=2 \hat{i}+3 \hat{j}+\hat{k}\) are coplanar and the projection of \(\vec{a}\) on the vector \(\vec{b}\) is \(\sqrt{54}\) units, then the sum of all possible values of \(\lambda+\mu\) is equal to

[JEE Main 2023, 29 Jan (Shift 1)]

Let \(\vec{u}=\hat{i}-\hat{j}-2 \hat{k}, \vec{v}=2 \hat{i}+\hat{j}-\hat{k}, \vec{v} \cdot \vec{w}=2\) and \(\vec{v} \times \vec{w}=\vec{u}+\lambda \vec{v}\). Then \(\vec{u} \cdot \vec{w}\) is equal to

[JEE Main 2023, 24 Jan (Shift 1)]

Let \(\vec{a}=\hat{i}+4 \hat{j}+2 \hat{k}, \vec{b}=3 \hat{i}-2 \hat{j}+7 \hat{k}\) and \(\vec{c}=2 \hat{i}-\hat{j}+4 \hat{k}\). If a vector \(\vec{d}\) satisfies \(\vec{d} \times \vec{b}=\vec{c} \times \vec{b}\) and \(\vec{d} \cdot \vec{a}=24\), then \(|\vec{d}|^2\) is equal to

[JEE Main 2023, 13 Apr (Shift 1)]

\(\text{ If }|\vec{a}|=2,|\vec{b}|=5\text{ and }|\vec{a}\times \vec{b}|=8\text{, then }|\vec{a}\cdot \vec{b}|\text{ is equal to: }\)

[JEE Main 2021, 25 Jul (Shift 2)]

If \(\vec{a}, \vec{b}, \vec{c}\) are three non-zero vectors and \(\hat{n}\) is a unit vector perpendicular to \(\vec{c}\) such that \(\vec{a}=\alpha \vec{b}-\hat{n},(\alpha \neq 0)\) and \(\vec{b} \cdot \vec{c}=12\), then \(|\vec{c} \times(\vec{a} \times \vec{b})|\) is equal to:

[JEE Main 2023, 30 Jan (Shift 1)]

Let \(\lambda \in Z , \vec{a}=\lambda \hat{i}+\hat{j}-\hat{k}\) and \(\vec{b}=3 \hat{i}-\hat{j}+2 \hat{k}\). Let \(\vec{c}\) be a vector such that \((\vec{a}+\vec{b}+\vec{c}) \times \vec{c}=\overrightarrow{0}, \vec{a} \cdot \vec{c}=-17\) and \(\vec{b} \cdot \vec{c}=-20\). Then \(|\vec{c} \times(\lambda \hat{i}+\hat{j}+\hat{k})|^2\) is equal to

[JEE Main 2023, 12 Apr (Shift 1)]

If \(\ \vec{a} \) is a vector of magnitude 50 , collinear with the vector \(\ \vec{b}=6 \hat{i}-8 \hat{j}-\frac{15}{2} \hat{k} \) and makes an acute angle with the positive direction of \(\ \mathbf{Z} \) - axis, then \(\ \vec{a} \) is equal to

For any vector \(\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}\), with \(10\left|a_i\right|<1, i=1,2,3\), consider the following statements:
(A) \(\max \left\{\left|a_1\right|,\left|a_2\right|,\left|a_3\right|\right\} \leq|\vec{a}|\)
(B) \(|\vec{a}| \leq 3 \max \left\{\left|a_1\right|,\left|a_2\right|,\left|a_3\right|\right\}\)

[JEE Main 2023, 11 Apr (Shift 1)]

Let \(\left|\vec{a}\right|=2,\left|\vec{b}\right|=3\) and the angle between the vectors \(\vec{a}\text{ and  }\vec{b}\text{  be   }\frac{\pi }{4}.\) Then \({\left|\left(\vec{a}+2\vec{b}\right)\times \left(2\vec{a}−3\vec{b}\right)\right|}^{2}\) is equal to

[JEE Main 2023, 13 Apr (Shift 2)]

If the points with position vectors \(6 \hat{i}+11 \hat{j}+11 \hat{k}\), \(\frac{9}{2} \hat{i}+\beta \hat{j}-8 \hat{k}, \alpha \hat{i}+10 \hat{j}+13 \hat{k}\) are collinear, then \((19 \alpha-6 \beta)^2\) is equal to

[JEE Main 2023, 8 Apr (Shift 1)]

If \(\vec{a}, \vec{b}\) and \(\vec{c}\) are three unit vectors satisfying \(|\vec{a}-\vec{b}|^2+|\vec{b}-\vec{c}|^2+|\vec{c}-\vec{a}|^2=9\), then \(|3 \vec{a}+3 \vec{b}+8 \vec{c}|\) equals

Let \(\vec{a}=2 \hat{i}+\hat{j}+\hat{k}\), and \(\vec{b}\) and \(\vec{c}\) be two nonzero vectors such that \(|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}|\) and \(\vec{b} \cdot \vec{c}=0\). Consider the following two statement:
(A) \(|\vec{a}+\lambda \vec{c}| \geq|\vec{a}|\) for all \(\lambda \in R\).
(B) \(\vec{a}\) and \(\vec{c}\) are always parallel. Then

[JEE Main 2023, 31 Jan (Shift 1)]

Let \(\vec{a}=2 \hat{i}+\hat{j}+\hat{k}\), and \(\vec{b}\) and \(\vec{c}\) be two nonzero vectors such that \(|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}|\) and \(\vec{b} \cdot \vec{c}=0\). Consider the following two statement:
(A) \(|\vec{a}+\lambda \vec{c}| \geq|\vec{a}|\) for all \(\lambda \in R\).
(B) \(\vec{a}\) and \(\vec{c}\) are always parallel

Then

[JEE Main 2023, 31 Jan (Shift 1)]

If \(\vec{r}\) and \(\vec{s}\) are non-zero constant vectors and the scalar \(b\) is chosen such that \(|\vec{r}+b \vec{s}|\) is minimum, then the value of \(|b \vec{s}|^2+|\vec{r}+b \vec{s}|^2\) is equal to

Let the position vectors of the points P, Q, R and S be \(\vec{a}=\hat{i}+2\hat{j}−5\hat{k},\) \(\vec{b}=3\hat{i}+6\hat{j}+3\hat{k},\) \(\vec{c}=\frac{17}{5}\hat{i}+\frac{16}{5}\hat{j}+7\hat{k}\) and \(\vec{d}=2\hat{i}+\hat{j}+\hat{k},\) respectively. Then which of the following statements is true?

Let \(\vec{\alpha}=4 \hat{i}+3 \hat{j}+5 \hat{k}\) and \(\vec{\beta}=\hat{i}+2 \hat{j}-4 \hat{k}\). Let \(\vec{\beta}_1\) be parallel to \(\vec{\alpha}\) and \(\vec{\beta}_2\) be perpendicular to \(\vec{\alpha}\). If \(\vec{\beta}=\vec{\beta}_1+\vec{\beta}_2\), then the value of \(5 \vec{\beta}_2 \cdot(\hat{i}+\hat{j}+\hat{k})\) is

[JEE Main 2023, 24 Jan (Shift 2)]

Let \(\vec{a}=5 \hat{i}-\hat{j}-3 \hat{k}\) and \(\vec{b}=\hat{i}+3 \hat{j}+5 \hat{k}\) be two vectors. Then which one of the following statements is TRUE?

[JEE Main 2023, 1 Feb (Shift 2)]

Let \(\vec{a}=\hat{i}+2 \hat{j}-3 \hat{k}\) and \(\vec{b}=2 \hat{i}-3 \hat{j}+5 \hat{k}\).If \(\vec{r} \times \vec{a}=\vec{b} \times \vec{r}, \vec{r} \cdot(\alpha \hat{i}+2 \hat{j}+\hat{k})=3\) and \(\vec{r} \cdot(2 \hat{i}+5 \hat{j}-\alpha \hat{k})\), \(=-1, \alpha \in R\) then the value of \(\alpha+|\vec{r}|^2\) is equal to:

[JEE Main 2021, 16 Mar (Shift 2)]

Let \(\overrightarrow{ a }=\hat{ i }+\hat{ j }+\hat{ k }\) and \(\overrightarrow{ b }=\hat{ j }-\hat{ k }\). If \(\overrightarrow{ c }\) is a vector such that \(\vec{a} \times \vec{c}=\vec{b}\) and \(\vec{a} \cdot \vec{c}=3\), then \(\vec{a} \cdot(\vec{b} \times \vec{c})\) is equal to

[JEE Main 2021, 26 Aug (Shift 1)]

Let \(\vec{a}=4 \hat{i}+3 \hat{j}\) and \(\vec{b}=3 \hat{i}-4 \hat{j}+5 \hat{k}\) and \(\vec{c}\) is a vector such that \(\vec{c} \cdot(\vec{a} \times \vec{b})+25=0, \vec{c} \cdot(\hat{i}+\hat{j}+\hat{k})=4\) and projection of \(\vec{c}\) on \(\vec{a}\) is 1 , then the projection of \(\vec{c}\) on \(\vec{b}\) equals:

[JEE Main 2023, 29 Jan (Shift 2)]

The vector \(\vec{a}=-\hat{i}+2 \hat{j}+\hat{k}\) is rotated through a right angle, passing through the \(y\)-axis in its way and the resulting vector is \(\vec{b}\). Then the projection of \(3 \vec{a}+\sqrt{2} \vec{b}\) on \(\vec{c}=5 \hat{i}+4 \hat{j}+3 \hat{k}\) is

In a triangle ABC, if \(|\overrightarrow{\mathrm{BC}}|=8,|\overrightarrow{\mathrm{CA}}|=7,|\overrightarrow{\mathrm{AB}}|=10\), then the projection of the vector \(\overrightarrow{\mathrm{AB}}\) on \(\overrightarrow{\mathrm{AC}}\) is equal

[JEE Main 2021, 18 Mar (Shift 2)]

Let \(\vec{a}\) and \(\vec{b}\) are two non-zero vectors perpendicular to each other and \(|\vec{a}|=|\vec{b}|\). If \(|\vec{a} \times \vec{b}|=|\vec{a}|\), then the angle between the vectors \((\vec{a}+\vec{b}+(\vec{a} \times \vec{b}))\) and \(\vec{a}\) is equal to:

[JEE Main 2021, 18 Mar (Shift 2)]

Let the vectors \( \vec{a}, \vec{b}, \vec{c} \) be such that \( |\vec{a}|=2,|\vec{b}|=4 \) and \( |\vec{c}|=4 \). If the projection of \( \vec{b} \) on \( \vec{a} \) is equal to the projection of \( \vec{c} \) on \( \vec{a} \) and \( \vec{b} \) is perpendicular to \( \vec{c} \), then the value of \( |\vec{a}+\vec{b}-\vec{c}| \) is_________.

[JEE Main 2020, 5 Sep (Shift 2)]

Let \(a, b\), and \(c\) be distinct positive numbers. If the vectors \(a \hat{i}+a \hat{j}+c \hat{k}, \hat{i}+\hat{k}\) and \(c \hat{i}+c \hat{j}+b \hat{k}\) are coplanar, then \(c\) is equal to:

[JEE Main 2021, 25 Jul (Shift 2)]

For what value of \(m\) are the vector \(2\hat{i}−3\hat{j}+4\hat{k},\hat{i}+2\hat{j}−\hat{k}\) and \(m\hat{i}−\hat{j}+2\hat{k}\) are coplanar?

Let \(\vec{a}=2 \hat{i}-3 \hat{j}+4 \hat{k}\) and \(\vec{b}=7 \hat{i}+\hat{j}-6 \hat{k}\). If \(\vec{r} \times \vec{a}=\vec{r} \times \vec{b}, \vec{r} \cdot(\hat{i}+2 \hat{j}+\hat{k})=-3\), then \(\vec{r} \cdot(2 \hat{i}-3 \hat{j}+\hat{k})\) is equal to:

Let \(\vec{a}=2\hat{i}+3\hat{j}+4\hat{k},\text{  }b=\hat{i}−2\hat{j}−2\hat{k}\) and \(\vec{c}=−\hat{i}+4\hat{j}+3\hat{k}.\) if \(\vec{d}\) is a vector perpendicular to both \(\vec{b}\text{ and  }\vec{c}\), and \(\vec{a}.\vec{d}=18,\) then \({\left|\vec{a}\times \vec{d}\right|}^{2}\) is equal to:

[JEE Main 2023, 6 Apr (Shift 1)]

If four distinct points with position vectors \(\vec{a}, \vec{b}, \vec{c}\) and \(\vec{d}\) are coplanar; then \([\vec{a} \ \vec{b} \ \vec{c}]\) is equal to

[JEE Main 2023, 11 Apr (Shift 2)]

Let : \(\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}, \vec{b}=\hat{i}-\hat{j}+2 \hat{k}\) and \(\vec{c}=5 \hat{i}-3 \hat{j}+3 \hat{k}\) be three vectors. If \(\vec{r}\) is a vector such that, \(\vec{r} \times \vec{b}=\vec{c} \times \vec{b}\) and \(\vec{r} \cdot \vec{a}=0\). Then \(25|\vec{r}|^2\) is equal to

[JEE Main 2023, 31 Jan (Shift 2)]

Let \(\vec{a}=\hat{i}+\hat{j}+2\hat{k}\) and \(\vec{b}=-\hat{i}+2\hat{j}+3\hat{k}\). Then the vector product \((\vec{a}+\vec{b})\times ((\vec{a}\times ((\vec{a}-\vec{b})\times \vec{b}))\times \vec{b})\) is equal to:

[JEE Main 2021, 27 Jul (Shift 1)]

If the vectors \(\vec{a}=\lambda \hat{i}+\mu \hat{j}+4\hat{k},\vec{b}=−2\hat{i}+4\hat{j}−2\hat{k}\text{ }\) and \(\vec{c}=2\hat{i}+3\hat{j}+\hat{k}\) are coplanar and the projection of \(\vec{a}\) on the vector \(\vec{b}\) is \(\sqrt{54}\) then the sum of all possible values of \(\lambda +\mu\) is equal to

[JEE Main 2023, 29 Jan (Shift 1)]

Let \(\vec{a}=\hat{i}+2 \hat{j}-3 \hat{k}\) and \(\vec{b}=2 \hat{i}-3 \hat{j}+5 \hat{k}\).
If \(\vec{r} \times \vec{a}=\vec{b} \times \vec{r}, \vec{r} \cdot(\alpha \hat{i}+2 \hat{j}+\hat{k})=3\) and \(\vec{r} \cdot(2 \hat{i}+5 \hat{j}-\alpha \hat{k})\), \(=-1, \alpha \in R\) then the value of \(\alpha+|\vec{r}|^2\) is equal to:

[JEE Main 2021, 16 Mar (Shift 2)]

Let the vectors \( (2+a+b) \hat{i}+(a+2 b+c) \hat{j}-(b+c) \hat{k}, \ (1+b) \hat{i}+2 b j-b \hat{k}\) and \((2+b) \hat{i}+2 b \hat{j}+(1-b) \hat{k}, a, b, c \in R\) be co-planar. Then which of the following is true?

Let \(a, b, c\) be three distinct real numbers, none equal to one. If the vectors \(a \hat{i}+\hat{j}+\hat{k}, \hat{i}+b \hat{j}+\hat{k}\) and \(\hat{i}+\hat{j}+c \hat{k}\) are coplanar, then \(\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}\) is equal to

[JEE Main 2023, 12 Apr (Shift 1)]

Let \(a,b\) and \(c\) be the distinct non-negative numbers. If the vectors \(a\hat{i}+a\hat{j}+c\hat{k},\text{ }\hat{i}\text{ }+\text{ }\hat{k}\text{ , }c\hat{i}\text{ }+\text{ }c\hat{j}+b\hat{k}\) lie on a plane, then which one of the following is correct?

Let \(\vec{a}=4\hat{i}+5\hat{j}−\hat{k}\), \(\vec{b}=\hat{i}−4\hat{j}+5\hat{k}\) and \(\vec{c}=3\hat{i}+\hat{j}−\hat{k}.\) Let \(\vec{d}\) be a vector which is perpendicular to both \(\vec{a}\) and \(\vec{b}\) and \(\vec{c}⋅\vec{d}=12.\) Then \((-\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}).(\vec{\mathrm{c}}\times \vec{\mathrm{d}})\) is equal to

[JEE Main 2023, 10 Apr (Shift 2)]

If \(|\vec{a}|=2,|\vec{b}|=5\) and \(|\vec{a} \times \vec{b}|=8\), then \(|\vec{a} \cdot \vec{b}|\) is equal to:

[JEE Main 2021, 25 Jul (Shift 2)]

If \(\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{a} \cdot \vec{b}=1\) and \(\vec{a} \times \vec{b}=\\hat{j}-\hat{k}\), then \(\vec{b}\) is

Let \(\lambda \in R , \vec{a}=\lambda \hat{i}+2 \hat{j}-3 \hat{k}, \vec{b}=\hat{i}-\lambda \hat{j}+2 \hat{k}\). If \(((\vec{a}+\vec{b}) \times(\vec{a} \times \vec{b})) \times(\vec{a}-\vec{b})=8 \hat{i}-40 \hat{j}-24 \hat{k}\), then \(|\lambda(\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})|^2\) is equal to

[JEE Main 2023, 30 Jan (Shift 2)]

Let O be the origin and the position vector of the point P be \(−\hat{i}−2\hat{j}+3\hat{k}.\) If the position vectors of the points A, B and C are \(−2\hat{i}+\hat{j}−3\hat{k},\) \(2\hat{i}+4\hat{j}−2\hat{k}\) and \(−4\hat{i}+2\hat{j}−\hat{k}\) respectively, then the projection of the vector \(\vec{OP}\)on a vector perpendicular to the vectors \(\vec{AB}\) and \(\vec{AC}\) is

[JEE Main 2023, 10 Apr (Shift 1)]

Let three vectors \( \vec{a}, \vec{b} \) and \( \vec{c} \) be such that \( \vec{a} \times \vec{b}=\vec{c}, \vec{b} \times \vec{c}=\vec{a} \) and \( |\vec{a}|=2 \). Then which one of the following is not true?

[JEE Main 2021, 22 Jul (Shift 2)]

Let \( \vec{a} \) and \( \vec{b} \) be two vectors such that \( |2 \vec{a}+3 \vec{b}|=|3 \vec{a}+\vec{b}| \) and the angle between \( \vec{a} \) and \( \vec{b} \) is \( 60^{\circ} \). If \( \frac{1}{8} \vec{a} \) is a unit vector, then \( |\vec{b}| \) is equal to:

If, \(\vec{a}=\hat{i}+2 \hat{k}, \vec{b}=\hat{i}+\hat{j}+\hat{k}, \vec{c}=7 \hat{i}-3 \hat{j}+4 \hat{k}\), \(\vec{r} \times \vec{b}+\vec{b} \times \vec{c}=0\) and \(\vec{r} \cdot \vec{a}=0\) then \(\vec{r} \cdot \vec{c}\) is equal to:

Let \(\vec{a}, \vec{b}, \vec{c}\) be three vectors mutually perpendicular to each other and have same magnitude. If a vector \(\vec{r}\) satisfies \(\vec{a} \times\{(\vec{r}-\vec{b}) \times \vec{a}\}+\vec{b} \times\{(\vec{r}-\vec{c}) \times \vec{b}\}+\vec{c} \times\{(\vec{r}-\vec{a}) \times \vec{c}\}=\overrightarrow{0}\) then \(\vec{r}\) is equal to

[JEE Main 2021, 31 Aug (Shift 2)]

Let the vectors \(\vec{a}, \vec{b}, \vec{c}\) represent three coterminous edges of a parallelopiped of volume \(V\). Then the volume of the parallelopiped, whose coterminous edges are represented by \(\vec{a}, \vec{b}+\vec{c}\) and \(\vec{a}+2 \vec{b}+3 \vec{c}\) is equal to

[JEE Main 2023, 6 Apr (Shift 2)]

Let \( \vec{a}, \vec{b}, \vec{c} \) be three vectors mutually perpendicular to each other and have same magnitude. If a vector \( \vec{r} \) satisfies.

\(\vec{a}\times {(\vec{r}-\vec{b})\times \vec{a}}+\vec{b}\times {(\vec{r}-\vec{c})\times \vec{b}}+\vec{c}\times {(\vec{r}-\vec{a})\times \vec{c}}=\vec{0},\)

then \( \vec{r} \) is equal to:

[JEE Main 2021, 31 Aug (Shift 2)]

Let \(\lambda \in \mathrm{R},\vec{\mathrm{a}}=\lambda \hat{\mathrm{i}}+2\hat{\mathrm{j}}-3\hat{\mathrm{k}},\vec{\mathrm{b}}=\hat{\mathrm{i}}-\lambda \hat{\mathrm{j}}+2\hat{\mathrm{k}}.\)

If \(((\vec{\mathrm{a}}+\vec{\mathrm{b}})\times (\vec{\mathrm{a}}\times \vec{\mathrm{b}}))\times (\vec{\mathrm{a}}-\vec{\mathrm{b}})=8\hat{\mathrm{i}}-40\hat{\mathrm{j}}-24\hat{\mathrm{k}}\) then \(|\lambda (\vec{\mathrm{a}}+\vec{\mathrm{b}})\times (\vec{\mathrm{a}}-\vec{\mathrm{b}}){|}^{2}\) is equal to

Let \(\vec{a}=2 \hat{i}+\hat{j}+\hat{k}\), and \(\vec{b}\) and \(\vec{c}\) be two nonzero vectors such that \(|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}|\) and \(\vec{b} \cdot \vec{c}=0\). Consider the following two statement:
(A) \(|\vec{a}+\lambda \vec{c}| \geq|\vec{a}|\) for all \(\lambda \in R\).
(B) \(\vec{a}\) and \(\vec{c}\) are always parallel. Then

Let \(O\) be the origin. Let \(\overrightarrow{O P}=x \hat{i}+y \hat{j}-\hat{k}\) and \(\overrightarrow{O Q}=-\hat{i}+2 \hat{j}+3 x \hat{k}, x, y \in R, x>0\), be such that \(|\overrightarrow{P Q}|=\sqrt{20}\) and the vector \(\overrightarrow{O P}\) is perpendicular to \(\overrightarrow{O Q}\). If \(\overrightarrow{O R}=3 \hat{i}+z \hat{j}-7 \hat{k}, z \in R\), is coplanar with \(\overrightarrow{O P}\) and \(\overrightarrow{O Q}\) , then the value of \(x^2+y^2+z^2\) is equal to:

[JEE Main 2021, 17 Mar (Shift 2)]

For any vector \(\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}\), with \(10\left|a_i\right|<1, i=1,2,3\), consider the following statements:
(A) : \(\max \left\{\left|a_1\right|,\left|a_2\right|,\left|a_3\right|\right\} \leq|\vec{a}|\)
(B) : \(|\vec{a}| \leq 3 \max \left\{\left|a_1\right|,\left|a_2\right|,\left|a_3\right|\right\}\)

[JEE Main 2023, 11 Apr (Shift 1)]

Let \(\vec{a}\) and \(\vec{b}\)be two vectors. Let \(|\vec{a}|=1,|\vec{b}|=4\) and \(\vec{a}⋅\vec{b}=2\).

If , \(\vec{c}=(2\vec{a}\times \vec{b})−3\vec{b}\) , then the value of \(\vec{b}⋅\vec{c}\)is

Let \(\vec{a}\) and \(\vec{b}\) be two vectors. Let \(|\vec{a}|=1,|\vec{b}|=4\) and \(\vec{a} \cdot \vec{b}=2\).If \(\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}\), then the value of \(\vec{b} \cdot \vec{c}\) is

Let the vectors \({\vec{u}}_{1}=\hat{i}+\hat{j}+a\hat{k},{\vec{u}}_{2}=\hat{i}+b\hat{j}+\hat{k}\) and \({\vec{u}}_{3}=c\hat{i}+\hat{j}+\hat{k}\) be coplanar. If the vectors \({\vec{v}}_{1}=(a+b)\hat{i}+c\hat{j}+c\hat{k},\) \({\vec{v}}_{2}=a\hat{i}+(b+c)\hat{j}+a\hat{k}\) and \({\vec{v}}_{3}=b\hat{i}+b\hat{j}+(c+a)\hat{k}\) are also coplanar, then 6(a + b + c) is equal to

[JEE Main 2023, 08 Apr (Shift 2)]

Let \(\vec{a}=2 \hat{i}+7 \hat{j}-\hat{k}, \vec{b}=3 \hat{i}+5 \hat{k}\) and \(\vec{c}=\hat{i}-\hat{j}+2 \hat{k}\). Let \(\vec{d}\) be a vector which is perpendicular to both \(\vec{a}\) and \(\vec{b}\), and \(\vec{c} \cdot \vec{d}=12\). Then \((-\hat{i}+\hat{j}-\hat{k}) \cdot(\vec{c} \times \vec{d})\) is equal to

[JEE Main 2023, 10 Apr (Shift 2)]

If \(\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{a} \cdot \vec{b}=1\) and \(\vec{a} \times \vec{b}=\vec{j}-\vec{k}\), then \(\vec{b}\) is

If the points with position vectors \(\alpha \hat{i}+10\hat{j}+13\hat{k},\text{     }6\hat{i}+11\hat{j}+11\hat{k}\), \(\frac{9}{2}\hat{i}+\beta \hat{j}−8\hat{k}\) are collinear, then (19α – 6β)² is equal to

[JEE Main 2023, 08 Apr (Shift 1)]

Let \(\lambda \in R , \vec{a}=\lambda \hat{i}+2 \hat{j}-3 \hat{k}, \vec{b}=\hat{i}-\lambda \hat{j}+2 \hat{k}\). If \(((\vec{a}+\vec{b}) \times(\vec{a} \times \vec{b})) \times(\vec{a}-\vec{b})=8 \hat{i}-40 \hat{j}-24 \hat{k}\), then \(|\lambda(\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})|^2\) is equal to

Let the position vectors of the points \(P, Q, R\) and \(S\) be \(\vec{a}=\hat{i}+2 \hat{j}-5 \hat{k}, \vec{b}=3 \hat{i}+6 \hat{j}+3 \hat{k}, \quad \vec{c}=\frac{17}{5} \hat{i}+\frac{16}{5} \hat{j}+7 \hat{k}\) and \(\vec{d}=2 \hat{i}+\hat{j}+\hat{k}\), respectively. Then which of the following statements is true?

[JEE Advanced 2023]

Let : \(\vec{a}=\hat{i}+2\hat{j}+3\hat{k},\vec{b}=\hat{i}−\hat{j}+2\hat{k}\) and \(\vec{c}=5\hat{i}−3\hat{j}+3\hat{k}\) be there vectors. If \(\vec{r}\) is a vector such that, \(\vec{r}\times \vec{b}=\vec{c}\times \vec{b}\) and \(\vec{r}⋅\vec{a}=0,\) then \(25{\left|\vec{r}\right|}^{2}\) is equal to

[JEE Main 2023, 30 Jan (Shift 2)]

Let a vector \(\vec{a}\) be coplanar with vectors \(\vec{b}=2\hat{i}+\hat{j}+\hat{k}\text{ and }\vec{c}=\hat{i}-\hat{j}+\hat{k}\text{. }\)If \(\vec{a}\) is perpendicular to \(\vec{d}=3\hat{i}+2\hat{j}+6\hat{k}\text{ and }|\vec{a}|=\sqrt{10}\) Then a possible value of \([\vec{a}\vec{b}\vec{c}]+[\vec{a}\vec{b}\vec{d}]+[\vec{a}\vec{c}\vec{d}]\) is equal to :

[JEE Main 2021, 22 Jul (Shift 2)]

Let \(S\) be the set of all \((\lambda, \mu)\) for which the vectors \(\lambda \hat{i}-\hat{j}+\hat{k}, \hat{i}+2 \hat{j}+\mu \hat{k}\) and \(3 \hat{i}-4 \hat{j}+5 \hat{k}\), where \(\lambda-\mu=5\), are coplanar, then \(\sum_{(\lambda, \mu) \in S} 80\left(\lambda^2+\mu^2\right)\) is equal to:

[JEE Main 2023, 15 Apr (Shift 1)]

If four distinct points with position vectors \(\vec{a}, \vec{b}, \vec{c}\) and \(\vec{d}\) are coplanar; then \([\vec{a} \vec{b} \vec{c}]\) is equal to

[JEE Main 2023, 11 Apr (Shift 2)]

Let \(\vec{a}\) and \(\vec{b}\) be two vectors. Let \(|\vec{a}|=1,|\vec{b}|=4\) and \(\vec{a} \cdot \vec{b}=2\).If \(\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}\), then the value of \(\vec{b} \cdot \vec{c}\) is

[JEE Main 2023, 30 Jan (Shift 2)]

Let \(\vec{a}=2\hat{i}−7\hat{j}+5\hat{k},\vec{b}=\hat{i}+\hat{k}\) and \(\vec{c}=\hat{i}+2\hat{j}−3\hat{k}\) be three given vectors. If \(\vec{r}\) is a vector such that \(\vec{r}\times \vec{a}=\vec{c}\times \vec{a}\) and \(\vec{r}⋅\vec{b}=0\), then \(|\vec{r}|\) is equal to:

[JEE Main 2023, 1 Feb (Shift 2)]

Let \(O\) be the origin. Let \(\overrightarrow{O P}=x \hat{i}+y \hat{j}-\hat{k}\) and \(O Q=-\hat{i}+2 \hat{j}+3 x \hat{k}, x, y \in R, x>0\), be such that \(|\overrightarrow{P Q}|=\sqrt{20}\) and the vector \(\overrightarrow{O P}\) is perpendicular to \(\overrightarrow{O Q}\). If \(\overrightarrow{O R}=3 \hat{i}+z \hat{j}-7 \hat{k}, z \in R\), is coplanar with \(\overrightarrow{O P}\) and \(\overrightarrow{O Q}\) , then the value of \(x^2+y^2+z^2\) is equal to:

[JEE Main 2021, 17 Mar (Shift 2)]

Let \(\vec{a}\) and \(\vec{b}\) are two non-zero vectors perpendicular to each other and \(|\vec{a}|=|\vec{b}|\). If \(|\vec{a}\times \vec{b}|=|\vec{a}|\), then the angle between the vectors \((\vec{a}+\vec{b}+(\vec{a}\times \vec{b}))\) and \(\vec{a}\) is equal to:

[JEE Main 2021, 18 Mar (Shift 2)]

Let \(x_0\) be the point of local maxima of \(f(x)=\vec{a} \cdot(\vec{b} \times \vec{c})\), where \(\vec{a}=x \hat{i}-2 \hat{j}+3 \hat{k}, \vec{b}=-2 \hat{i}+x \hat{j}-\hat{k}, \vec{c}=7 \hat{i}-2 \hat{j}+x \hat{k} \). Then the value of \(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}\) at \(x=x_0\) is

[JEE Main 2020, 4 Sep (Shift 1)]

Let \(\vec{a}=-\hat{i}-\hat{j}+\hat{k},\vec{a}.\vec{b}=1\) and \(\vec{a}\times \vec{b}=\hat{i}-\hat{j}\)Then \(\vec{a}-6\vec{b}\) is equal to

Let the position vectors of the points \(A, B, C\) and \(D\) be \(5 \hat{i}+5 \hat{j}+2 \lambda \hat{k}, \hat{i}+2 \hat{j}+3 \hat{k},-2 \hat{i}+\lambda \hat{j}+4 \hat{k}\) and \(-\hat{i}+5 \hat{j}+6 \hat{k}\). Let the set \(S=\{\lambda \in R\) : The points \(A, B, C\) and \(D\) are coplanar \(\}\). Then \(\sum_{\lambda \in S}(\lambda+2)^2\) is equal to

[JEE Main 2023, 6 Apr (Shift 1)]