Three Dimensional Geometry

Direction ratios of a vector parallel to line \(\frac{x-1}{2}=-y=\frac{2 z+1}{6}\) are :

Direction ratios of a vector parallel to line \(\frac{x-1}{2}=-y=\frac{2 z+1}{6}\) are :

Assertion (A) : A line in space cannot be drawn perpendicular to \(\mathrm{x}, \mathrm{y}\) and \(z\) axes simultaneously.

Reason \((R)\) : For any line making angles, \(\alpha, \beta, \gamma\) with the positive directions of \(x, y\) and \(z\) axes respectively, then \(\cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1\).

The equation of a line parallel to the vector \(3\hat{i}+\hat{j}+2\hat{k}\) and passing through the point \((4,-3,7)\) is :

The distance of point \(\mathrm{P}(\mathrm{a}, \mathrm{b}, \mathrm{c})\) from y -axis is :

The distance of point \(\mathrm{P}(\mathrm{a}, \mathrm{b}, \mathrm{c})\) from y -axis is :

The coordinates of the foot of the perpendicular drawn from the point \((0,1,2)\) on the \(x\)-axis are given by :

If a line makes an angle of \(30^{\circ}\) with the positive direction of \(x\)-axis, \(120^{\circ}\) with the positive direction of \(y\)-axis, then the angle which it makes with the positive direction of \(z\)-axis is :

The angle which the line \(\frac{x}{1}=\frac{\mathrm{y}}{-1}=\frac{\mathrm{z}}{0}\) makes with the positive direction of Y -axis is :

The line \(x=1+5\mu ,y=-5+\mu ,z=-6-3\mu\) passes through which of the following point?

The equation of a line parallel to the vector \(3\hat{i}+\hat{j}+2\hat{k}\) and passing through the point \((4,-3,7)\) is :

If a line makes an angle of \(30^{\circ}\) with the positive direction of \(x\)-axis, \(120^{\circ}\) with the positive direction of \(y\)-axis, then the angle which it makes with the positive direction of \(z\)-axis is :

The Cartesian equation of the line passing through the point \(\left(1,-3,2\right)\) and parallel to the line : \(\vec{\mathrm{r}}=\left(2+\lambda \right)\hat{\mathrm{i}}+\lambda \hat{\mathrm{j}}+\left(2\lambda -1\right)\hat{\mathrm{k}}\) is

The Cartesian equation of the line passing through the point \((1,-3,2)\) and parallel to the line : \(\vec{\mathrm{r}}=(2+\lambda )\hat{\mathrm{i}}+\lambda \hat{\mathrm{j}}+(2\lambda -1)\hat{\mathrm{k}}\) is

The angle which the line \(\frac{x}{1}=\frac{\mathrm{y}}{-1}=\frac{\mathrm{z}}{0}\) makes with the positive direction of Y -axis is :

If the direction cosines of a line are \(\sqrt{3} \mathrm{k}, \sqrt{3} \mathrm{k}, \sqrt{3} \mathrm{k}\), then the value of k is :

If a line makes angles of \(\frac{3\pi }{4},\frac{\pi }{3}\) and \(\theta\) with the positive directions of \(x, y\) and \(z\)-axis respectively, then \(\theta\) is

If the direction cosines of a line are \(\sqrt{3} \mathrm{k}, \sqrt{3} \mathrm{k}, \sqrt{3} \mathrm{k}\), then the value of k is :

The coordinates of the foot of the perpendicular drawn from the point \((0,1,2)\) on the \(x\)-axis are given by :

The line \(x=1+5\mu ,y=-5+\mu ,z=-6-3\mu\) passes through which of the following point?

If a line makes angles of \(\frac{3\pi }{4},\frac{\pi }{3}\) and \(\theta\) with the positive directions of \(x, y\) and \(z\)-axis respectively, then \(\theta\) is

Assertion (A) : A line in space cannot be drawn perpendicular to \(\mathrm{x}, \mathrm{y}\) and z axes simultaneously.

Reason \((R)\) : For any line making angles, \(\alpha, \beta, \gamma\) with the positive directions of \(x, y\) and \(z\) axes respectively, \(\cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1\).

The shortest distance between the lines \(\frac{\mathrm{x}-3}{2}=\frac{\mathrm{y}-2}{3}=\frac{\mathrm{z}-1}{-1}\) and \(\frac{\mathrm{x}+3}{2}=\frac{\mathrm{y}-6}{1}=\frac{\mathrm{z}-5}{3}\) is :

The shortest distance between the lines \(\frac{x+2}{1}=\frac{y}{-2}=\frac{z-5}{2}\) and \(\frac{x-4}{1}=\frac{y-1}{2}=\frac{z+3}{0}\) is

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

The equation of the plane with intercepts of 2,3 and 4 on the x, y and z-axes respectively is -