$A$ point on the $XOZ$-plane divides the line segment joining the points $(5, -3, -2)$ and $(1, 2, -2)$ at:

Find a vector $\vec{r}$ of magnitude $3 \sqrt{2}$ units which makes an angle of $\frac{\pi}{4}$ and $\frac{\pi}{2}$ with $y$ and $z$-axes,respectively.

If the direction cosines $l, m, n$ of two lines satisfy the relations $l+m+n=0$ and $lm=0$,then the angle between those two lines is

$A$ line $AB$ in three-dimensional space makes angles $45^{\circ}$ and $120^{\circ}$ with the positive $x$-axis and the positive $y$-axis respectively. If $AB$ makes an acute angle $\theta$ with the positive $z$-axis,then $\theta$ equals

$\text{Assertion (A)}$: The direction ratios of line $L_1$ are $2, 5, 7$ and those of line $L_2$ are $\frac{4}{\sqrt{19}}, \frac{10}{\sqrt{19}}, \frac{14}{\sqrt{19}}$. The lines $L_1, L_2$ are parallel.
$\text{Reason (R)}$: The direction ratios of a line $L_1$ are $a_1, b_1, c_1$ and those of another line $L_2$ are $a_2, b_2, c_2$. The lines $L_1$ and $L_2$ are parallel if $a_1 a_2+b_1 b_2+c_1 c_2=0$.
The correct option among the following is

If the angle between the lines whose direction cosines are $\left(-\frac{2}{\sqrt{21}}, \frac{C}{\sqrt{21}}, \frac{1}{\sqrt{21}}\right)$ and $\left(\frac{3}{\sqrt{54}}, \frac{3}{\sqrt{54}}, -\frac{6}{\sqrt{54}}\right)$ is $\frac{\pi}{2}$,then the value of $C$ is