Question

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

• A. $47$
• B. $-25$
• C. $0$
• D. $25$

Answer 1

The correct option is B $-25$

A] Given U+V+W = 0

Taking dot product with U,V, W :-

U.U+V.U+W.U = 0---(1)

U.V+V.V+W.W = 0---(2)

U.W+V.W+W.W = 0---(3)

adding (1),(2),(3)

$2\left(\right(U.V)+(U.W)+(V.W\left)\right)=-({\left|U\right|}^2+{\left|V\right|}^2+{\left|W\right|}^2)$

= -(9+16+25)

= -50

$\therefore$ U.V+V.W+V.W=-25