For a unit vector $\vec{a}$,if $(\vec{x}-\vec{a}) \cdot (\vec{x}+\vec{a}) = 15$,then find $|\vec{x}|$.

If the $p^{th}$,$q^{th}$,and $r^{th}$ terms of a geometric progression are $a$,$b$,and $c$ respectively,then find the angle between the vectors $\vec{u} = (\log a)\hat{i} + (\log b)\hat{j} + (\log c)\hat{k}$ and $\vec{v} = (q - r)\hat{i} + (r - p)\hat{j} + (p - q)\hat{k}$.

$P$ is the circumcentre of $\triangle ABC$. If the position vectors of $A, B, C$ and $P$ are $\bar{a}, \bar{b}, \bar{c}$ and $\frac{\bar{a}+\bar{b}+\bar{c}}{4}$ respectively,then the position vector of the orthocentre of this triangle is

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

The vector $2i + j - k$ is perpendicular to $i - 4j + \lambda k,$ if $\lambda = $

If the adjacent sides of a rectangle are $\bar{a}=5\bar{m}-3\bar{n}$,$\bar{b}=-\bar{m}-2\bar{n}$ and the adjacent sides of another rectangle are $\bar{c}=-4\bar{m}-\bar{n}$,$\bar{d}=-\bar{m}+\bar{n}$,then the angle between the vectors $\bar{x}=\frac{\bar{a}+\bar{c}+\bar{d}}{3}$ and $\bar{y}=\frac{\bar{c}+\bar{d}}{5}$ is