The vectors $\vec{AB} = 3\hat{i} - 2\hat{j} + 2\hat{k}$ and $\vec{BC} = \hat{i} - 2\hat{k}$ are the adjacent sides of a parallelogram. The angle between its diagonals is

If $A=(-2,2,3), B=(3,2,2), C=(4,-3,5)$ and $D=(7,-5,-1)$,then the projection of $\overline{AB}$ on $\overline{CD}$ is

For $p > 0$,a vector $\vec{v}_{2} = 2 \hat{i} + (p + 1) \hat{j}$ is obtained by rotating the vector $\vec{v}_{1} = \sqrt{3} p \hat{i} + \hat{j}$ by an angle $\theta$ about the origin in the counter-clockwise direction. If $\tan \theta = \frac{(\alpha \sqrt{3} - 2)}{4 \sqrt{3} + 3}$,then the value of $\alpha$ is equal to $....$

Let $a$ and $b$ be two unit vectors inclined at an angle $\theta$,then $\sin(\theta/2)$ is equal to

$ABC$ is a right-angled triangle in which $\max \{AB, BC, AC\} = BC$. If the position vectors of $B$ and $C$ are respectively $3\hat{i}-2\hat{j}+\hat{k}$ and $5\hat{i}+\hat{j}-3\hat{k}$,then find the value of $AB \cdot AC + BA \cdot BC + CA \cdot CB$.

If $a$ and $b$ are mutually perpendicular vectors,then $(a + b)^2 = $