The vector $c$ directed along the internal bisector of the angle between the vectors $a = 7i - 4j - 4k$ and $b = -2i - j + 2k$ with $|c| = 5\sqrt{6}$ is

$\hat{i} \cdot (\hat{j} \times \hat{k}) + \hat{j} \cdot (\hat{i} \times \hat{k}) + \hat{k} \cdot (\hat{i} \times \hat{j}) + \hat{j} \cdot (\hat{j} \times \hat{k}) = $ . . . . . . .

$A$ tetrahedron has vertices at $O(0, 0, 0)$,$A(1, 2, 1)$,$B(2, 1, 3)$,and $C(-1, 1, 2)$. The angle between the faces $OAB$ and $ABC$ is:

Given $\vec{a}=3 \hat{i}-\hat{j}$,$\vec{b}=2 \hat{i}+\hat{j}-3 \hat{k}$ and $\vec{b}=\overrightarrow{b_1}+\overrightarrow{b_2}$,where $\overrightarrow{b_1}$ is parallel to $\vec{a}$ and $\overrightarrow{b_2}$ is perpendicular to $\vec{a}$,then $\overrightarrow{b_2}$ is equal to

The cosine of the angle $A$ of the triangle with vertices $A(1, -1, 2)$,$B(6, 11, 2)$,and $C(1, 2, 6)$ is:

The projection of the line segment joining the points $P(1, -1, 3)$ and $Q(2, -4, 11)$ on the line joining the points $A(-1, 2, 3)$ and $B(3, -2, 10)$ is