Find the projection of the vector $\hat{i}+3 \hat{j}+7 \hat{k}$ on the vector $7 \hat{i}-\hat{j}+8 \hat{k}$.

If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors such that $\vec{a}$ is perpendicular to both $\vec{b}$ and $\vec{c}$, and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{2 \pi}{3}$, then $|\vec{a}+3 \vec{b}-4 \vec{c}|^2=$

If $\vec{a} = \hat{i} - 2\hat{j} + 2\hat{k}$ and $\vec{b} = 2\hat{i} - 3\hat{j} + \hat{k}$, then the component of $\vec{b}$ perpendicular to $\vec{a}$ is

Let $(p \wedge q)$ denote the angle between vectors $p$ and $q$. If $a + b + c = 0$, $|a| = 7$, $|b| = 5$, and $|c| = 3$, find the correct relation.

Let $ABCD$ be a parallelogram such that $\vec{AB} = \vec{q}$ and $\vec{AD} = \vec{p}$,and $\angle BAD$ is an acute angle. If $\vec{r}$ is the vector that coincides with the altitude directed from the vertex $B$ to the side $AD$,then $\vec{r}$ is given by:

The position vectors of points $A, B, C$ are given by $2\hat{i} - \hat{j} + \hat{k}$,$\hat{i} - 3\hat{j} - 5\hat{k}$,and $a\hat{i} - 3\hat{j} + \hat{k}$ respectively. If these points form a right-angled triangle with $\angle C = \pi/2$,find the value of $a$.