$A$ unit vector perpendicular to the plane of $a = 2i - 6j - 3k$ and $b = 4i + 3j - k$ is

If $\bar{a}, \bar{b}, \bar{c}$ are non-coplanar unit vectors such that $\bar{a} \times(\bar{b} \times \bar{c})=\frac{(\bar{b}+\bar{c})}{\sqrt{2}}$,then the angle between $\bar{a}$ and $\bar{b}$ is

Let $\hat{\alpha}, \hat{\beta}, \hat{\gamma}$ be three unit vectors such that $\hat{\alpha} \times (\hat{\beta} \times \hat{\gamma}) = \frac{1}{2}(\hat{\beta} + \hat{\gamma})$. If $\hat{\beta}$ is not parallel to $\hat{\gamma}$, then the angle between $\hat{\alpha}$ and $\hat{\beta}$ is

Find a vector of magnitude $6,$ which is perpendicular to both the vectors $\vec{a} = 2 \hat{i}-\hat{j}+2 \hat{k}$ and $\vec{b} = 4 \hat{i}-\hat{j}+3 \hat{k}$.

Let $u$ and $v$ be two non-zero vectors. Then the magnitude of the cross product $u \times v$ is always

If $a = 2 \hat{i} + 3 \hat{j} - \hat{k}$,$b = \hat{i} + 2 \hat{j} - 5 \hat{k}$,and $c = 3 \hat{i} + 5 \hat{j} - \hat{k}$,then a vector perpendicular to $a$ and in the plane containing $b$ and $c$ is: