The unit vector perpendicular to both $i + j$ and $j + k$ is

$r \times a = b \times a;\,\,r \times b = a \times b;\,\,a \ne 0;\,\,b \ne 0;\,\,a \ne \lambda b;\,\,a$ is not perpendicular to $b,$ then $r = $

If $a = 2i + 2j - k$ and $b = 6i - 3j + 2k,$ then the value of $a \times b$ is

If the position vectors of three points $A, B, C$ are $\hat{i} + \hat{j} + \hat{k}$,$2\hat{i} + 3\hat{j} - 4\hat{k}$,and $7\hat{i} + 4\hat{j} + 9\hat{k}$ respectively,then find the unit vector perpendicular to the plane of triangle $ABC$.

Let $\vec{a}$ and $\vec{b}$ be two vectors of length $\sqrt{2}$ such that $|\vec{a} + \vec{b}| = \sqrt{5}$. If $\vec{c} = \vec{a} + 2\vec{b} + 2(\vec{a} \times \vec{b})$,then $|\vec{c}|$ is

If $(2 \hat{i} + 6 \hat{j} + 27 \hat{k}) \times (\hat{i} + \lambda \hat{j} + \mu \hat{k}) = 0$,then $\lambda + \mu =$ . . . . . . .