If $|a \times b|^2 + |a \cdot b|^2 = 36$ and $|a| = 3$,then $|b|$ is equal to

The orthogonal projection of vector $a$ on vector $b$ is given by:

If $a = t^2 \hat{i} + e^t \hat{j} + \hat{k}$ and $b = 2 \hat{i} + t^2 \hat{j} + \log t \hat{k}$,and $f(t) = a \cdot b$,then $f^{\prime}(1)$ is equal to

Let $\vec{u}$ and $\vec{v}$ be two non-zero vectors with the intermediate angle $45^{\circ}$. Then $|\vec{u} \times \vec{v}|=$

If $P=(0,1,2)$, $Q=(4,-2,1)$, and $O=(0,0,0)$, then $\angle POQ$ is equal to

If the vectors $\vec{a} = \hat{i} - \hat{j} + 2\hat{k}$,$\vec{b} = 2\hat{i} + 4\hat{j} + \hat{k}$,and $\vec{c} = \lambda\hat{i} + \hat{j} + \mu\hat{k}$ are mutually perpendicular,then $(\lambda, \mu) = .......$