Question

Let $u , v$ and $w$ be vectors such that $u + v + w = 0 .$ If $| u |=3,| v |=4$ and $| w |=5$ then $u \cdot v + v \cdot w + w \cdot u$ is equal to

• A. 0
• B. -25
• C. 25
• D. 50

Answer 1

Answer: (B)

-25

Answer 2

Given, $| u |=3,| v |=4$ and $| w |=5$
Also, $u + v + w = 0$
On squaring both sides, we get
$| u |^{2}+| v |^{2}+| w |^{2}+2( u \cdot v + v \cdot w + w \cdot u )=0$
$\Rightarrow 3^{2}+4^{2}+5^{2}+2( u \cdot v + v \cdot w + w \cdot u )=0$
$\Rightarrow 9+16+25+2( u \cdot v + v \cdot w + w \cdot u )=0$
$\Rightarrow u \cdot v + v \cdot w + w \cdot u =-\frac{50}{2}=-25$