Let $S$ be the mirror image of the point $Q(1,3,4)$ with respect to the plane $2 x-y+z+3=0$ and let $R(3,5, \gamma)$ be a point of...

Let $\vec{a}=\hat{i}+5 \hat{j}+\alpha \hat{k}, \vec{b}=\hat{i}+3 \hat{j}+\beta \hat{k}$ and $\vec{c}=-\hat{i}+2 \hat{j}+3 \hat{k}$ be three vectors such that, $|\vec{b} \times \vec{c}|=5 \sqrt{3}$ and $\vec{a}$ is perpendicular to...

The equation of the line passing through $(-4,3,1)$, parallel to the plane $x+2 y-z-5=0$ and intersecting the line $\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z-2}{-1}$ is :

The length of the perpendicular from the point ( $2,-$ 1,4) on the straight line, $\frac{x+3}{10}=\frac{y-2}{-7}=\frac{z}{1}$ is

Let $Q$ be the foot of the perpendicular from the point $P(7,-2,13)$ on the plane containing the lines $\frac{x+1}{6}=\frac{y-1}{7}=\frac{z-3}{8}$ and $\frac{x-1}{3}=\frac{y-2}{5}=\frac{z-3}{7}$. Then $(P Q)^2$, is...

The magnitude of the projection of the vector $2 \hat{i}+3 \hat{j}+\hat{k}$ on the vector perpendicular to the plane containing the vectors $\hat{i}+\hat{j}+\hat{k}$ and $...

If the projection of the vector $\hat{i}+2 \hat{j}+\hat{k}$ on the sum of the two vectors $2 \hat{i}+4 \hat{j}-5 \hat{k}$ and $-\lambda \hat{i}+2 \hat{j}+3 \hat{k}$ is...

The equation of a plane containing the line of intersection of the planes $2 x-y-4=0$ and $y+2 z -4=0$ and passing through the point $(1,1,0)$...

The plane through the intersection of the planes x + y + z = 1 and 2x + 3y – z + 4 = 0...

Let $P$ be the plane passing through the point $(1,2,3)$ and the line of intersection of the planes $\vec{r} \cdot(\hat{i}+\hat{j}+4 \hat{k})=16$ and $\vec{r} \cdot(-\hat{i}+\hat{j}+\hat{k})=6$. Then...