Tag: three dimensional geometry - ii

If $\theta$ denotes the acute angle between the line $\bar{r} = (\bar{i} + 2\bar{j} - \bar{k}) + \lambda  (\bar{i} - \bar{j} + \bar{k})$ and the plane $\bar{r} = (2\bar{i} - \bar{j} + \bar{k}) = 4$, then $\sin \theta + \sqrt 2 \cos \theta$

Let $\vec {AB}=\hat {i}-\hat {j}+\hat {k}$ be rotated about $A$ along the plane $3x-y-2z=5$ by an angle $\cos^{-1}\dfrac {\sqrt {2}}{3}$ so that the point $B$ reaches the point $C$, then the vector representing $AC$ may be

Gives the line $\displaystyle L:\frac { x-1 }{ 3 } =\frac { y+1 }{ 2 } =\frac { z-3 }{ -1 } $ and the plane $\pi :x-2y=0$. Of the following assertions, the only one that is always true is:

Consider a plane $x + y - z = 1$ and the point $A(1, 2, -3)$. A line $L$ has the equation $x = 1 + 3r$, $y = 2 - r$, $z = 3 + 4r$

If the angle between the line $x=\dfrac{y-1}{2}=\dfrac{z-3}{\lambda}$ and the plane $x+2y+3z=4$ is $\cos ^{ -1 }{ \left( \sqrt { 5/14 }  \right)  } $ then $\lambda$=

Consider plane containing line $\dfrac{x+1}{-3} = \dfrac{y-3}{z} = \dfrac{z+2}{-1}$ and passing through the point $(1, -1, 0)$. The angle made by the plane with x-axis is

Consider plane containing line $\dfrac{x+1}{-3} = \dfrac{y-3}{2} = \dfrac{z+2}{-1}$ and passing through the point $(1, -1, 0)$ . The angle made by the plane with x-axis is 

If the plane $2x-3y+6z-11=0$ makes an angle $\sin^{-1}(k)$ with x-axis, then $k$ is equal to:

The angle between the line $\displaystyle x = y = z$ and the plane $\displaystyle 4x - 3y + 5z = 2$ is

Given the line $\displaystyle L:\frac { x-1 }{ 3 } =\frac { y+1 }{ 2 } =\frac { z-3 }{ -1 } $ and the plane $\pi :x-2y=0$. Of the following assertion, the only one that is always true is