If the angle between the lines whose direction ratios are $2, -1, 2$ and $a, 3, 5$ is $45^\circ$,then $a =$

If the projections of a line segment on the $x, y$ and $z-$ axes in $3-$ dimensional space are $2, 3$ and $6$ respectively,then the length of the line segment is

If a variable line in two adjacent positions has direction cosines $l, m, n$ and $l+\delta l, m+\delta m, n+\delta n,$ show that the small angle $\delta \theta$ between the two positions is given by $\delta \theta^{2}=\delta l^{2}+\delta m^{2}+\delta n^{2}$.

If the projections of the line segment $\overrightarrow{AB}$ on $XY$,$YZ$,and $ZX$ planes are $\sqrt{15}$,$\sqrt{46}$,and $7$ respectively,then the projection of $\overrightarrow{AB}$ on the $y$-axis is

If the direction cosines of two lines are such that $l+m+n=0$ and $l^2+m^2-n^2=0$, then the angle between them is

$A$ unit vector $\hat{a}$ makes angles $\frac{\pi}{3}$ with $\hat{i}$,$\frac{\pi}{4}$ with $\hat{j}$,and $\theta \in (0, \pi)$ with $\hat{k}$. Then a value of $\theta$ is: