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

If $$a = i + j + k,\overrightarrow b = 4i + 3j + 4k$$ and $$c = i + \alpha j + \beta k$$ are linearly dependent vectors and $$\left| c \right| = \sqrt 3 ,$$ then

For three vectors $$u,v,w$$ which of the following expression is not equal to any of the remaining three?

Let $$\overrightarrow u ,\overrightarrow v $$ and $$\overrightarrow w $$ be vectors such that $$\overrightarrow u + \overrightarrow v + \overrightarrow w = 0.$$ If $$\left| {\overrightarrow u } \right| = 3,\left| {\overrightarrow v } \right| = 4$$ and $$\left| {\overrightarrow w } \right| = 5,$$ then $$\overrightarrow u .\overrightarrow v + \overrightarrow v .\overrightarrow w + \overrightarrow w .\overrightarrow u $$ is