$A$ vector of length $3$ perpendicular to each of the vectors $3\,i + j - 4\,k$ and $6\,i + 5\,j - 2\,k$ is

If $P=3 \hat{i}+5 \hat{j}-\hat{k}$ and $Q=\hat{i}+2 \hat{j}+3 \hat{k}$ are two sides of a triangle,then its area is equal to . . . . . . sq units.

If $\hat{u}$ and $\hat{v}$ are unit vectors and $\theta$ is the acute angle between them,then for what value of $\theta$ is $2\hat{u} \times 3\hat{v}$ a unit vector?

Find the unit vector perpendicular to the plane containing the vectors $\vec{a} = 2\hat{i} - 6\hat{j} - 3\hat{k}$ and $\vec{b} = 4\hat{i} + 3\hat{j} - \hat{k}$.

If the vector $\vec{b} = 3\hat{j} + 4\hat{k}$ is written as the sum of a vector $\vec{b_1}$,parallel to $\vec{a} = \hat{i} + \hat{j}$ and a vector $\vec{b_2}$,perpendicular to $\vec{a}$,then $\vec{b_1} \times \vec{b_2}$ is equal to

If $a$ and $b$ are unit vectors such that $a \times b$ is also a unit vector,then the angle between $a$ and $b$ is