The projection of vector $\vec{a}$ along vector $\vec{b}$ is:

If $(2, -1, 2)$ and $(K, 3, 5)$ are the triads of direction ratios of two lines and the angle between them is $45^{\circ}$,then the value of $K$ is

For non-zero vectors $a$ and $b$, if $|a+b| < |a-b|$, then $a$ and $b$ are

Let $\overrightarrow{OA}$ and $\overrightarrow{OB}$ be two sides of a triangle. The median $\overrightarrow{AM}$ is perpendicular to the angle bisector $\overrightarrow{OL}$ and $|\overrightarrow{AM}|:|\overrightarrow{OL}|=1:2$. The angle between $\overrightarrow{OA}$ and $\overrightarrow{OB}$ is

The position vectors of the vertices of $\triangle ABC$ are $4\hat{i} - 2\hat{j}$,$\hat{i} + 4\hat{j} - 3\hat{k}$,and $-\hat{i} + 5\hat{j} + \hat{k}$ respectively. Then,$m \angle ABC = $

If $p = i - 2j + 3k$ and $q = 3i + j + 2k,$ then a vector along $r$ which is a linear combination of $p$ and $q$ and also perpendicular to $q$ is