If $\vec{a} = \hat{i} + \hat{j}$ and $\vec{b} = \hat{i} - \hat{k}$, then the point of intersection of the lines $\vec{r} \times \vec{a} = \vec{b} \times \vec{a}$ and $\vec{r} \times \vec{b} = \vec{a} \times \vec{b}$ is

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

If $\bar{a}, \bar{b}, \bar{c}, \bar{d}$ are the position vectors of the points $A, B, C, D$ respectively such that $3 \bar{a}-\bar{b}+2 \bar{c}-4 \bar{d}=\overline{0}$,then the position vector of the point of intersection of the line segments $AC$ and $BD$ is

For what value of $m$ is the angle between the vectors $2\bar{i} - m\bar{j} + 3m\bar{k}$ and $(1 + m)\bar{i} - 2m\bar{j} + \bar{k}$ acute?

The vectors $a, b$ and $c$ are of the same length and taken pairwise,they form equal angles. If $a = i + j$ and $b = j + k,$ then the co-ordinates of $c$ are

Let $u, v, w$ be vectors such that $|u| = 1, |v| = 2, |w| = 3$. If the projection of $v$ along $u$ is equal to the projection of $w$ along $u$,and $v$ and $w$ are perpendicular to each other,then $|u - v + w|$ equals: