Let $\mathbf{a}=2 \mathbf{i}+\mathbf{j}+\mathbf{k}, \mathbf{b}=\mathbf{i}+2 \mathbf{j}-\mathbf{k}$ and $a$ unit vector $\mathbf{c}$ be coplanar. If $\mathbf{c}$ is perpendicular to $\mathbf{a}$, then c equals to

$\widehat u$ and $\widehat v$ are two non-collinear unit vectors such that $\left| {{{\widehat u + \widehat v} \over 2} + \widehat u \times \widehat v} \right| = 1$. Then the value of $|\widehat u \times \widehat v|$ is equal to

The points with position vectors $10\widehat i + 3\widehat j$, $12\widehat i - 5\widehat j$ and $a\widehat i + 11\widehat j$ are collinear, if a is

Let a, b, c be vectors of lengths 3, 4, 5 respectively and a be perpendicular to (b + c), b to (c + a) and c to (a + b), then the value of (a + b + c) is

For non-zero vectors a, b, c; |(a $\times$ b) . c| = |a| |b| |c| holds if and only if

If a and b are two vectors such that | a | = 1, | b | = 4 a . b = 2. If c = (2a $\times$ b) $-$ 3b, then angle between b and c

If $a = - \widehat i + \widehat j + \widehat k$ and $b = 2\widehat i + \widehat k$, then find z component of a vector r, which is coplanar with a and b, r . b = 0 and r . a = 7.