Let $P_1: \bar{r} .(2 \hat{i}+\hat{j}-3 \hat{k})=4$ be a plane. Let $P_2$ be another plane which passes through the points $(2,-3,2)(2$, $-2,-3)$ and $(1,-4,2)$. If the...

Let $d$ be the distance between the foot of perpendiculars of the points $P(1,2-1)$ and $Q(2,-1,3)$ on the plane $-x+y+z=1$. Then $d^2$ is equal to...

Let $P$ be the plane passing through the intersection of the planes $\vec{r} \cdot(\hat{i}+3 \hat{j}-\hat{k})=5$ and $\vec{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})=3$, and the point $(2,1,-2)$. Let the...

Let the mirror image of the point $(a, b, c)$ with respect to the plane $3 x-4 y+12 z+19=0$ be $(\alpha-6, \beta, \gamma)$. If $...

If $\vec{a}=2 \hat{i}+\hat{j}+3 \hat{k}, \vec{b}=3 \hat{i}+3 \hat{j}+\hat{k}$ and $\vec{c}=c_1 \hat{i}+c_2 \hat{j}+c_3 \hat{k}$ are coplanar vectors and $\vec{a} \cdot \vec{c}=5, \vec{b} \perp \vec{c}$, then $122\left(c_1+c_2\right. +\mathrm{c}_3$...

Let the plane $P: \vec{r} \cdot \vec{a}=d$ contain the line of intersection of two planes $\vec{r} \cdot(\hat{i}+3 \hat{j}-\hat{k})=6$ and $\vec{r} \cdot(-6 \hat{i}+5 \hat{j}-\hat{k})=7$. If the...

The acute angle between the planes $\mathrm{P}_1$ and $\mathrm{P}_2$, when $\mathrm{P}_1$ and $\mathrm{P}_2$ are the planes passing through the intersection of the planes $...

If two distinct point $Q, R$ lie on the line of intersection of the planes $-x+2 y-z=0$ and $3 x-5 y+2 z=0$ and $...

Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$. Then the number of vectors $\overrightarrow{\mathrm{b}}$ such that $\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{a}}$ and $|\overrightarrow{\mathrm{b}}| \in\{1,2, \ldots \ldots, 10\}$ is

If two straight lines whose direction cosines are given by the relations $I+m-n=0,3 I^2+m^2+c n l=0$ are parallel, then the positive value of $c$ is: