$A$ 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:

Scalar projection of the line segment joining the points $A(-2,0,3)$ and $B(1,4,2)$ on the line whose direction ratios are $6,-2,3$ is

Show that the area of the parallelogram whose diagonals are given by $\vec{a}$ and $\vec{b}$ is $\frac{|\vec{a} \times \vec{b}|}{2}$. Also,find the area of the parallelogram whose diagonals are $2 \hat{i}-\hat{j}+\hat{k}$ and $\hat{i}+3 \hat{j}-\hat{k}$.

If $A, B, C, D$ are the points $(2, 3, -1), (3, 5, -3), (1, 2, 3), (3, 5, 7)$ respectively,then the angle between $AB$ and $CD$ is

If $|a+b|=|a-b|$,then

If $\theta$ is the angle between two vectors $\vec{a}$ and $\vec{b}$ such that $|\vec{a}|=7$, $|\vec{b}|=1$ and $|\vec{a} \times \vec{b}|^2 = k^2 - (\vec{a} \cdot \vec{b})^2$, then the values of $k$ and $\theta$ are