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.