The equation of the plane containing the straight line $\frac{x}{2}=\frac{y}{3}=\frac{z}{4}$ and perpendicular to the plane containing the straight lines $\frac{x}{3}=\frac{y}{4}=\frac{z}{2}$ and $\frac{x}{4}=\frac{y}{2}=\frac{z}{3}$ is

Let $\ddot{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\sqrt{2} \hat{\mathrm{k}}, \ddot{\mathrm{b}}=\mathrm{b}_1 \hat{\mathrm{i}}+\mathrm{b}_2 \hat{\mathrm{j}}+\sqrt{2} \hat{\mathrm{k}}$ and $\ddot{\mathrm{c}}=5 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\sqrt{2} \hat{\mathrm{k}}$ be three vectors such that the projection vector of $\stackrel{a}{b}$ on $\stackrel{a}{a}$ is $\stackrel{n}{a}$....

Let the vectors $\vec{a}, \vec{b}, \vec{c}$, be such that $|\vec{a}|=2,|\vec{b}|=4$ and $|\vec{c}|=4$ and $|\vec{c}|=4$. If the projection of $\vec{b}$ on $\vec{a}$ is equal to the...

The direction ratios of normal to the plane through the points $(0,-1,0)$ and $(0,0,1)$ and making an angle $\frac{\pi}{4}$ with the plane $y-z+5=0$ are

The shortest distance between the lines
$\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}$ and $\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{2}$ is

Let the volume of a parallelopiped whose coterminous edges are given by $\overrightarrow{\mathrm{u}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\lambda \hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{v}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{w}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ be 1 cu. Unit. If...

Let $\vec{a}=\hat{i}+2 \hat{j}+4 \hat{k}, \vec{b}=\hat{i}+\lambda \hat{j}+4 \hat{k}$ and $\overrightarrow{\mathrm{c}}=2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+\left(\lambda^2-1\right) \hat{\mathrm{k}}$ be coplanar vectors. Then the non-zero vector $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}$ is :

The plane containing the line $\frac{x-3}{2}=\frac{y+2}{-1}=\frac{z-1}{3}$ and also containing its projection on the plane $2 x+3 y-z=$ 5 , contains which one of the following...

The distance of the point having position vector $-\hat{i}+2 \hat{j}+6 \hat{k}$ from the straight line passing through the point ( $2,3,-4$ ) and parallel to...

If the plane $2 x-y+2 z+3=0$ has the distances $\frac{1}{3}$ and $\frac{2}{3}$ units from the planes $4 x-2 y+4 z+\lambda=0$ and $2 x-y+2 z+\mu=0$, respectively,...