If the moduli of $a$ and $b$ are equal and the angle between them is $120^\circ$ and $a \cdot b = -8$,then $|a|$ is equal to

If $a \perp b$ and $(a+b) \perp (a+mb)$,then $m$ is equal to

The cosine of the angle between any two diagonals of a cube is

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 $PQR$ be a triangle such that $\overrightarrow{PQ}=-2\hat{i}-\hat{j}+2\hat{k}$ and $\overrightarrow{PR}=a\hat{i}+b\hat{j}-4\hat{k}$, where $a, b \in \mathbb{Z}$. Let $S$ be the point on $QR$, which is equidistant from the lines $PQ$ and $PR$. If $|\overrightarrow{PR}|=9$ and $\overrightarrow{PS}=\hat{i}-7\hat{j}+2\hat{k}$, then the value of $3a-4b$ is . . . . . . .

If $\bar{a} = \bar{i} + \sqrt{11} \bar{j} - 2 \bar{k}$ and $\bar{b} = \bar{i} + \sqrt{11} \bar{j} - 10 \bar{k}$ are two vectors, then the component of $\bar{b}$ perpendicular to $\bar{a}$ is: