If $a = i + j$ and $b = 2i - k$ are two vectors, then the point of intersection of two lines $r \times a = b \times a$ and $r \times b = a \times b$ is

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 $\vec{v} = (q - r)\hat{i} + (r - p)\hat{j} + (p - q)\hat{k}$.

If $a \cdot b = a \cdot c$,$a \times b = a \times c$ and $a \neq 0$,then

$a$ and $b$ are unit vectors such that $a+2b$ is also a unit vector. If $\theta$ is the angle between $a$ and $b$,then $\sin \theta + \cos^3 \theta + \tan^5 \theta$ is equal to

Let $\overline{a}=2 \hat{i}+\hat{j}-2 \hat{k}$ and $\overline{b}=\hat{i}+\hat{j}$. Let $\overline{c}$ be a vector such that $|\bar{c}-\bar{a}|=3$ and $|(\bar{a} \times \bar{b}) \times \bar{c}|=3$ and the angle between $\overline{c}$ and $\overline{a} \times \overline{b}$ is $30^{\circ}$,then $\overline{a} \cdot \overline{c}$ is equal to

If $\overline{a}=\hat{i}-2 \hat{j}+3 \hat{k}$ and $\overline{b}=2 \hat{i}+3 \hat{j}-\hat{k}$ are two vectors,then the angle between the vectors $3 \bar{a}+5 \bar{b}$ and $5 \bar{a}+3 \bar{b}$ is