If $\bar{a}$ and $\bar{b}$ are mutually perpendicular unit vectors,then $(3\bar{a}+2\bar{b}) \cdot (5\bar{a}-6\bar{b}) = $

Let $P, Q, R$ and $S$ be the points on the plane with position vectors $-2 \hat{i}-\hat{j}, 4 \hat{i}, 3 \hat{i}+3 \hat{j}$ and $-3 \hat{i}+2 \hat{j}$ respectively. The quadrilateral $PQRS$ must be a

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 $AD=$

If the vectors $\vec{BC} = 2\hat{i} + \hat{j} + \hat{k}$ and $\vec{CD} = \hat{i} + 2\hat{j} - 2\hat{k}$ represent two adjacent sides of a parallelogram $ABCD$ and $\theta$ is the angle between its diagonals $\vec{AC}$ and $\vec{BD}$, then $\tan \theta =$

Let $\theta$ denote the angle between vectors $\vec{a}$ and $\vec{b}$. If $\vec{a}=2 \hat{i}+3 \hat{j}+6 \hat{k}$,$\vec{a} \cdot \vec{b}=4$ and $\theta=\cos ^{-1}\left(\frac{4}{21}\right)$,then $\vec{a}+\vec{b}$ is:

Let $a, b$ and $c$ be vectors with magnitudes $3, 4$ and $5$ respectively and $a + b + c = 0$. Then the value of $a \cdot b + b \cdot c + c \cdot a$ is: