If $3$ vectors $a, b, c$ are such that $a \neq 0$ and $a \times b = 2(a \times c)$,$|a| = 1$,$|c| = 1$,$|b| = 4$,and the angle between $b$ and $c$ is $\cos^{-1}\left(\frac{1}{4}\right)$,and $b - 2c = \lambda a$,then $\lambda = $

The scalars $l$ and $m$ such that $la + mb = c,$ where $a, b$ and $c$ are given vectors,are equal to

Let the unit vectors $a$ and $b$ be perpendicular and the unit vector $c$ be inclined at an angle $\theta$ to both $a$ and $b$. If $c = \alpha a + \beta b + \gamma (a \times b)$,then

The point of intersection of $r \times a = b \times a$ and $r \times b = a \times b$,where $a = i + j$ and $b = 2i - k$ is

Let $(x, y) \in (R \times R)$ and $\vec{a} = x \hat{i} + 2 \hat{j} - \hat{k}$, $\vec{b} = 6 \hat{i} - y \hat{j} + 2 \hat{k}$ be two vectors. If $|\vec{a} \times \vec{b}|^2 + |\vec{a} \cdot \vec{b}|^2 = f(x) g(y)$, then $f(x) + g(y) - 46 = 0$ represents:

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