The angle between the vectors $\bar{a} = 6 \hat{i} + 2 \hat{j} - 8 \hat{k}$ and $\bar{b} = 4 \hat{i} - 4 \hat{j} + 2 \hat{k}$ is . . . . . . .

If $\vec{a} + \vec{b} + \vec{c} = \vec{0}$,$|\vec{a}| = 3$,$|\vec{b}| = 5$,and $|\vec{c}| = 7$,find the angle between $\vec{a}$ and $\vec{b}$.

$A$ force of magnitude $6$ acts along the vector $(9, 6, -2)$ and passes through a point $A(4, -1, -7)$. The moment of the force about the point $O(1, -3, 2)$ is

The vectors $a, b$ and $c$ are of the same length and taken pairwise,they form equal angles. If $a = i + j$ and $b = j + k,$ then the co-ordinates of $c$ are

If in a right-angled triangle $ABC$,the hypotenuse $|\overrightarrow{AB}| = p$,then $\overrightarrow{AB} \cdot \overrightarrow{AC} + \overrightarrow{BC} \cdot \overrightarrow{BA} + \overrightarrow{CA} \cdot \overrightarrow{CB} = $

Given $ABC$ is an equilateral triangle of side length $1$ unit and $P$ be any arbitrary point on the circumcircle of triangle $ABC,$ then $|\vec{PA}|^2+|\vec{PB}|^2+|\vec{PC}|^2$ is equal to