If $a$ and $b$ are unlike vectors,then $a \cdot b =$

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 the position vectors of the vertices of a triangle are $2\hat{i} - \hat{j} + \hat{k}$,$\hat{i} - 3\hat{j} - 5\hat{k}$,and $3\hat{i} - 4\hat{j} - 4\hat{k}$,then the triangle is:

If in a right-angled triangle $ABC$,the hypotenuse $|\overrightarrow{AB}| = p$,then $\overrightarrow{AB} \cdot \overrightarrow{AC} + \overrightarrow{BC} \cdot \overrightarrow{BA} + \overrightarrow{CA} \cdot \overrightarrow{CB} = $

Let $O$ be the origin and $PQR$ be an arbitrary triangle. If a point $S$ satisfies the condition $\overrightarrow{OP} \cdot \overrightarrow{OQ} + \overrightarrow{OR} \cdot \overrightarrow{OS} = \overrightarrow{OR} \cdot \overrightarrow{OP} + \overrightarrow{OQ} \cdot \overrightarrow{OS} = \overrightarrow{OQ} \cdot \overrightarrow{OR} + \overrightarrow{OP} \cdot \overrightarrow{OS}$,then the point $S$ is the:

If $\vec{a}$ and $\vec{b}$ are two vectors such that $\vec{a}=2 \hat{i}+2 \hat{j}+p \hat{k}$, $|\vec{b}|=7$, $\vec{a} \cdot \vec{b}=4$ and $|\vec{a} \times \vec{b}|=5 \sqrt{17}$, then $p=$