Vector Algebra

\(\text{ If }\vec{a}\text{ and }\vec{b}\text{ are two unit vectors, then }\vec{a}\cdot \vec{b}=...........\)

\((\vec{i}+3\vec{j}-2\vec{k})\times (-\vec{i}+3\vec{k})=\)

Let \(\vec{p}\) and \(\vec{q}\) be two unit vectors and \(\alpha\) be the angle between them. Then \((\vec{p}+\vec{q})\) will be a unit vector for what value of \(\alpha\) ?

\(\text{ If }\vec{a}\text{ and }\vec{b}\text{ are two unit vectors, then }\vec{a}\cdot \vec{b}=...........\)

Let \(\vec{a}=2\hat{i}−\hat{j}+\hat{k}\) and \(\vec{b}=\lambda \hat{j}+2\hat{k},\text{  }\lambda \in Z\) be the two vectors. Let \(\vec{c}=\vec{a}\times \vec{b}\) and \(\vec{d}\) be the vector of magnitude \(2\) in \(yz\)-plane. If \(\left|\vec{c}\right|=\sqrt{53},\) then the maximum possible value of \((\vec{c} \cdot \vec{d})^2\) is equal to:

[JEE Main 2026, 22 Jan (Shift 2)]

\(\left|\vec{i}-\vec{j}-\vec{k}\right|=\)

For any two vectors \(\vec{a}\) and \(\vec{b}\), which of the following statements is always true?

For any two vectors \(\vec{a}\) and \(\vec{b}\), which of the following statements is always true?

The projection vector of vector \(\vec{a}\) on vector \(\vec{b}\) is

The vector with terminal point \(\mathrm{A}(2,-3,5)\) and initial point \(\mathrm{B}(3,-4,7)\) is :

The unit vector perpendicular to both vectors \(\hat{i}+\hat{k}\) and \(\hat{i}-\hat{k}\) is:

The projection vector of \(\vec{a}\) on \(\vec{b}\) is

\(\vec{j}.\vec{j}=\)

\((4\vec{i}+3\vec{j}+3\vec{k}).(6\vec{i}-4\vec{j}+\vec{k})=\)

The position vectors of points P and Q are \(\vec{p}\) and \(\vec{q}\) respectively. The point R divides line segment PQ in the ratio 3: 1 and S is the mid-point of line segment PR. The position vector of S is :

\((\vec{i}+3\vec{j}-2\vec{k})\times (-\vec{i}+3\vec{k})=\)

Let \(\theta\) be the angle between two unit vectors \(\hat{\mathrm{a}}\) and \(\hat{\mathrm{b}} \operatorname{such}\) that \(\sin \theta=\frac{3}{5}\). Then, \(\hat{a} \cdot \hat{b}\) is equal to :

Let \(\theta\) be the angle between two unit vectors \(\hat{\mathrm{a}}\) and \(\hat{\mathrm{b}} \operatorname{such}\) that \(\sin \theta=\frac{3}{5}\). Then, \(\hat{a} \cdot \hat{b}\) is equal to :

The unit vector perpendicular to both vectors \(\hat{i}+\hat{k}\) and \(\hat{i}-\hat{k}\) is:

The vector with terminal point \(\mathrm{A}(2,-3,5)\) and initial point \(\mathrm{B}(3,-4,7)\) is :

\((4\vec{i}+3\vec{j}+3\vec{k}).(6\vec{i}-4\vec{j}+\vec{k})=\)

\(\vec{j}.\vec{j}=\)

Let \(\vec{p}\) and \(\vec{q}\) be two unit vectors and \(\alpha\) be the angle between them. Then \((\vec{p}+\vec{q})\) will be a unit vector for what value of \(\alpha\) ?

The position vectors of points \(P\) and \(Q\) are \(\vec{p}\) and \(\vec{q}\) respectively. The point \(R\) divides line segment \(PQ\) in the ratio \(3:1\) and \(S\) is the mid-point of line segment \(PR\). The position vector of \(S\) is :

\(\left|\vec{i}-\vec{j}-\vec{k}\right|=\)

If \(\vec{a}\) and \(\vec{b}\) are two non-zero vectors, then \(\vec{a}\cdot \vec{b}=-|\vec{a}||\vec{b}|\), if -

\(1+{\cot }^{2}\theta =\)

If the magnitude of two vectors \(\vec{a}\) and \(\vec{b}\) are \(\sqrt{3}\) and 2 respectively and \(\vec{a}\cdot \vec{b}=\sqrt{6}\), then the angle between \(\vec{a}\) and \(\vec{b}\) is -

The value of \(\hat{i}\cdot (\hat{j}\times \hat{k})+\hat{j}\cdot (\hat{i}\times \hat{k})+\hat{k}\cdot (\hat{i}\times \hat{j})\) is -

The value of \(\hat{i}\cdot (\hat{j}\times \hat{k})+\hat{j}\cdot (\hat{i}\times \hat{k})+\hat{k}\cdot (\hat{i}\times \hat{j})\) is:

Direction : Two statements are given one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the following options :
Assertion (A) : The vectors

\(\vec{\mathrm{a}}=6\hat{\mathrm{i}}+2\hat{\mathrm{j}}-8\hat{\mathrm{k}}\\ \vec{\mathrm{b}}=10\hat{\mathrm{i}}-2\hat{\mathrm{j}}-6\hat{\mathrm{k}}\\ \vec{\mathrm{c}}=4\hat{\mathrm{i}}-4\hat{\mathrm{j}}+2\hat{\mathrm{k}}\)

represent the sides of a right angled triangle.
Reason (R) : Three non-zero vectors of which none of two are collinear forms a triangle if their resultant is zero vector or sum of any two vectors is equal to the third.