Let three vectors $\vec{a}, \vec{b}$ and $\vec{c}$ be such that $\vec{c}$ is coplanar with $\vec{a}$ and, $\vec{b}, \vec{a} \cdot \vec{c}=7$ and $\vec{b}$ is perpendicular to...

In a triangle ABC , if $|\overrightarrow{\mathrm{BC}}|=3,|\overrightarrow{\mathrm{CA}}|=5$ and $|\overrightarrow{\mathrm{BA}}|=7$, then the projection of the vector $\overrightarrow{\mathrm{BA}}$ on $\overrightarrow{\mathrm{BC}}$ is equal to

If the distance of the point $(1,-2,3)$ from the plane $x+2 y-3 z+10=0$ measured parallel to the line, $\frac{x-1}{3}=\frac{2-y}{m}=\frac{z+3}{1}$ is $\sqrt{\frac{7}{2}}$, then the value of...

Let $\vec{a}=\hat{i}+2 \hat{j}-3 \hat{k}$ and $\vec{b}=2 \hat{i}-3 \hat{j}+5 \hat{k}$ If $\vec{r} \times \vec{a}=\vec{b} \times \vec{r} \cdot(a \hat{i}+2 \hat{j}+\hat{k})=3$ and $...

If the foot of the perpendicular from point $(4,3,8)$ on the line $\mathrm{L}_1: \frac{\mathrm{x}-\mathrm{a}}{\ell}=\frac{\mathrm{y}-2}{3}=\frac{\mathrm{z}-\mathrm{b}}{4}, \ell \neq 0$ is $(3,5$, 7), then the shortest distance between...

Consider the line $L$ given by the equation $\frac{x-3}{2}=\frac{y-1}{1}=\frac{z-2}{1}$. Let $Q$ be the mirror image of the point $(2,3$, -1) with respect to $L$. Let...

If $(x, y, z)$ be an arbitrary point lying on a plane $P$ which passes through the point $(42,0,0),(0,42,0)$ and ( $0,0,42$ ), then the...

The lines $x=a y-1=z-2$ and $x=3 y-2=b z-2,(a b \neq 0)$ are coplanar, if .

The equation of the plane which contains the y-axis and passes through the point (1, 2, 3) is :

Let $\vec{a}=2 \hat{i}-3 \hat{j}+4 \hat{k}$ and $\vec{b}=7 \hat{i}+\hat{j}-6 \hat{k}$ If $\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}})=-3$ then $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+\hat{\mathrm{k}})$ is equal...