If $\vec{a}$ is a unit vector and $(\vec{x}-\vec{a}) \cdot (\vec{x}+\vec{a})=8$,then find $|\vec{x}|$.

The unit vector in $ZOX$ plane, making angles $45^{\circ}$ and $60^{\circ}$ respectively with $\vec{\alpha}=2 \hat{i}+2 \hat{j}-\hat{k}$ and $\vec{\beta}=\hat{j}-\hat{k}$ is

For three unit vectors $\vec{a}, \vec{b}, \vec{c}$ satisfying $|\vec{a}-\vec{b}|^{2}+|\vec{b}-\vec{c}|^{2}+|\vec{c}-\vec{a}|^{2}=9$ and $|2\vec{a}+k\vec{b}+k\vec{c}|=3$, the positive value of $k$ is:

$ABC$ is a right-angled triangle in which $\max \{AB, BC, AC\} = BC$. If the position vectors of $B$ and $C$ are respectively $3\hat{i}-2\hat{j}+\hat{k}$ and $5\hat{i}+\hat{j}-3\hat{k}$,then find the value of $AB \cdot AC + BA \cdot BC + CA \cdot CB$.

Let $\vec{p}$ and $\vec{q}$ be the position vectors of points $P$ and $Q$ respectively,with respect to the origin $O$,and let $|\vec{p}|=p, |\vec{q}|=q$. The points $R$ and $S$ divide the line segment $PQ$ internally and externally in the ratio $2:3$ respectively. If $\vec{OR}$ and $\vec{OS}$ are perpendicular,then:

If $\overrightarrow{a} \cdot \hat{i} = \overrightarrow{a} \cdot (\hat{i} + \hat{j}) = \overrightarrow{a} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1$,then $\overrightarrow{a}$ is equal to