The number of vectors of unit length perpendicular to vectors $a = (1, 1, 0)$ and $b = (0, 1, 1)$ is

If $(2 \hat{i} + 6 \hat{j} + 27 \hat{k}) \times (\hat{i} + \lambda \hat{j} + \mu \hat{k}) = \vec{0}$,then $\lambda$ and $\mu$ are respectively:

If the equation of a straight line passing through a point with position vector $a$ and parallel to the vector $b$ is $r = a + t\,b$,where $t$ is a parameter,then what is the perpendicular distance from this line to a point with position vector $c$?

Let $\bar{a}$ and $\bar{b}$ be two vectors such that $|\bar{a}|=1$,$|\bar{b}|=4$,and $\bar{a} \cdot \bar{b}=2$. If $\bar{c}=(2 \bar{a} \times \bar{b})-3 \bar{b}$,then the angle between $\bar{b}$ and $\bar{c}$ is

Let $\bar{a}=4 \bar{i}+5 \bar{j}-\bar{k}$,$\bar{b}=\bar{i}-4 \bar{j}+5 \bar{k}$,$\bar{c}=3 \bar{i}+\bar{j}-\bar{k}$ and let $\bar{\alpha}$ be a vector perpendicular to both $\bar{a}$ and $\bar{b}$ such that $\bar{\alpha} \cdot \bar{c}=63$. Then $\bar{\alpha}=$

Find $|a \times b|^2$,if $|a|=2, |b|=3$ and the angle between $a$ and $b$ is $\theta = \frac{\pi}{6}$.