Question

If a, b, c are unit vectors such that a + b + c = 0, then find the value of a.b + b.c + c.a

Answer 1

Given, |a| = |b| = |c| = 1 and a + b + c = 0

We have (a + b + c). (a + b + c) = 0

$\Rightarrow a.(a+b+c)+b.(a+b+c)+c.(a+b+c)=0$

$\Rightarrow a.a+a.b+a.c+b.a+b.b+b.c+c.a+c.b+c.c=0$

$\Rightarrow |a{|}^2+|b{|}^2+|c{|}^2+2(a.b+b.c+c.a)=0$

$(\because a.a=|a{|}^{2}anda.b=b.a)$

$\Rightarrow 1+1+1+2(a.b+b.c+c.a)=0$ ($\because |a|=|b|=|c|=1$

$\Rightarrow 3+2(a.b+b.c+c.a)=0$

$\Rightarrow a.b+b.c+c.a=-\frac{3}{2}$