Question
Let $$\vec u,\,\vec v$$ and $$\vec w$$ be vectors such that $$\vec u + \vec v + \vec w = 0.$$ If $$\left| {\vec u} \right| = 3,\,\left| {\vec v} \right| = 4$$ and $$\left| {\vec w} \right| = 5,$$ then $$\vec u.\vec v + \vec v.\vec w + \,\vec w.\vec u$$ is :
A.
$$47$$
B.
$$ - 25$$
C.
$$0$$
D.
$$25$$
Answer :
$$ - 25$$
Solution :
$$\eqalign{
& \because \vec u + \vec v + \vec w = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\therefore {\left| {\vec u + \vec v + \vec w} \right|^2} = 0 \cr
& \Rightarrow {\left| {\vec u} \right|^2} + \,{\left| {\vec v} \right|^2} + \,{\left| {\vec w} \right|^2} + 2\left( {\vec u.\vec v + \vec v.\vec w + \,\vec w.\vec u} \right) = 0 \cr
& \Rightarrow 9 + 16 + 25 + 2\left( {\vec u.\vec v + \vec v.\vec w + \,\vec w.\vec u} \right) = 0 \cr
& \Rightarrow \left( {\vec u.\vec v + \vec v.\vec w + \,\vec w.\vec u} \right) = - 25 \cr} $$