Let a straight line $L$ pass through the point $P(2,-1,3)$ and be perpendicular to the lines $\frac{x-1}{2}=\frac{y+1}{1}=\frac{z-3}{-2}$ and $\frac{x-3}{1}=\frac{y-2}{3}=\frac{z+2}{4}$. If the line L intersects the...

If the components of $\overrightarrow{\mathbf{a}}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$ along and perpendicular to $\overrightarrow{\mathbf{b}}=3 \hat{i}+\hat{j}-\hat{k}$ respectively, are $\frac{16}{11}(3 \hat{i}+\hat{j}-\hat{k})$ and $\frac{1}{11}(-4 \hat{i}-5 \hat{j}-17 \hat{k})$, then $\alpha^{2}+\beta^{2}+\gamma^{2}$...

If the shortest distance between the lines $\frac{{x - 1}}{2} = \frac{{y - 2}}{3} = \frac{{z - 3}}{4}$ and $...

Let the angle $\theta ,0 < \theta < \frac{\pi }{2}$ between two-unit vectors $\hat a$ and $\hat b$ be $...

Let $\mathrm{A}, \mathrm{B}, \mathrm{C}$ be three points in $\sqrt{3} \hat{i}+\hat{j}, \hat{i}+\sqrt{3} \hat{j}$ and $\mathrm{a} \hat{i}+(1-\mathrm{a}) \hat{j}$ respectively with respect to the origin O . If...

The square of the distance of the point $\left(\frac{15}{7}, \frac{32}{7}, 7\right)$ from the line $\frac{x+1}{3}=\frac{y+3}{5}=\frac{z+5}{7}$ in the direction of the vector $\hat{i}+4 \hat{j}+7 \hat{k}$ is:

Let the line L pass through (1,1,1) and intersect the lines $\frac{{x - 1}}{2} = \frac{{y + 1}}{3} = \frac{{z - 1}}{4}$ and $...

Let the shortest distance between the lines $\frac{{x - 3}}{3} = \frac{{y - \alpha }}{{ - 1}} = \frac{{z - 3}}{1}$ and $...

Let A and B be two distinct points on the line $L:\frac{{x - 6}}{3} = \frac{{y - 7}}{2} = \frac{{z - 7}}{{ - 2}}$. Both...

Consider two vectors $\vec u = 3\hat i - \hat j$ and $...