If $\overrightarrow{a} \cdot \hat{i}=4$,then $(\overrightarrow{a} \times \hat{j}) \cdot(2 \hat{j}-3 \hat{k})$ is equal to

If $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}|=|\vec{b}|=\sqrt{14}$ and $\vec{a} \cdot \vec{b}=-7$,then $\frac{|\vec{a} \times \vec{b}|}{|\vec{a} \cdot \vec{b}|}=$

For vectors $\vec{a}, \vec{b}$ and $\vec{c}$,if $\vec{a}+\vec{b}+\vec{c}=\vec{0}$ and $|\vec{a}|=2, |\vec{b}|=3, |\vec{c}|=5$,then the value of $\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$ is . . . . . . .

If $a\hat{i} + 6\hat{j} - \hat{k}$ and $7\hat{i} - 3\hat{j} + 17\hat{k}$ are perpendicular vectors,then what is the value of $a$?

If $a=\hat{i}+\hat{j}+t \hat{k}$ and $b=\hat{i}+2 \hat{j}+3 \hat{k}$,then the values of $t$ for which $(a+b)$ and $(a-b)$ are perpendicular are:

$ABCD$ is a parallelogram. The position vectors of $A$ and $C$ are respectively $3\hat{i} + 3\hat{j} + 5\hat{k}$ and $\hat{i} - 5\hat{j} - 5\hat{k}$. If $M$ is the midpoint of the diagonal $DB$,then the magnitude of the projection of $\vec{OM}$ on $\vec{OC}$,where $O$ is the origin,is