The angle between the vectors $(2i + 6j + 3k)$ and $(12i - 4j + 3k)$ is

If $P=(0,1,2), Q=(4,-2,1)$ and $O=(0,0,0)$,then $\angle POQ=$

If $A(4,7,8)$,$B(2,3,4)$,and $C(2,5,7)$ are the position vectors of the vertices of a triangle $ABC$ and if the internal bisector of $\angle A$ meets $BC$ at $D$,then $AD=$

Let $\vec{a} = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}$ with $a_{i} > 0$ for $i = 1, 2, 3$ be a vector that makes equal angles with the coordinate axes $OX$,$OY$,and $OZ$. Also,let the projection of $\vec{a}$ on the vector $3 \hat{i} + 4 \hat{j}$ be $7$. Let $\vec{b}$ be a vector obtained by rotating $\vec{a}$ by $90^{\circ}$. If $\vec{a}$,$\vec{b}$,and the $x$-axis are coplanar,then the projection of vector $\vec{b}$ on $3 \hat{i} + 4 \hat{j}$ is equal to

If vectors $\bar{a}=2 \hat{i}+2 \hat{j}+3 \hat{k}$,$\bar{b}=-\hat{i}+2 \hat{j}+\hat{k}$,and $\bar{c}=-3 \hat{i}+\hat{j}+2 \hat{k}$ are such that $\bar{a}+\lambda \bar{b}$ is perpendicular to $\bar{c}$,then $\lambda=$

If the vectors $\hat{i}+3 \hat{j}+4 \hat{k}$ and $\lambda \hat{i}-4 \hat{j}+\hat{k}$ are orthogonal to each other,then $\lambda$ is equal to