Question

If $a,b,c$ are vectors such that $a+b+c=0$ and $|a|=7,|b|=5,|c|=3$, then the angle between $c$ and $b$ is

• A. $\pi /3$
• B. $\pi /6$
• C. $\pi /4$
• D. $\pi$
• E. 0

Answer 1

Answer: (A)

We have,
$a+b+c=0$
$\Rightarrow b+c=-a$
$\Rightarrow |b+c|=|-a|$
$\Rightarrow |b+c|=|a|$
$\Rightarrow |b+c{|}^2=|a{|}^2$
$\Rightarrow (b+c)\cdot (b+c)=|a{|}^2$
$\Rightarrow |b{|}^2+|c{|}^2+2|b||c|\cos \theta =|a{|}^2$
$\Rightarrow (5{)}^2+(3{)}^2+2\times 5\times 3\cos \theta =(7{)}^2$
$\Rightarrow 25+9+30\cos \theta =49$
$\Rightarrow 30\cos \theta =15$
$\Rightarrow \cos \theta =\frac{1}{2}$
$\Rightarrow \theta =60^{\circ }$ or $\pi /3$
$\therefore$ Angle between $b$ and $c$ is $\frac{\pi }{3}$.