Question

If $a+b+c=0$ and $|a|=5,|b|=3$ and $|c|=7$ then angle between $a$ and $b$ is

• A. $\frac{\pi }{2}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{4}$
• D. $\frac{\pi }{6}$

Answer 1

The correct option is B $\frac{\pi }{3}$

Given, $a+b+c=0$
and $|a|=5,|b|=3,|c|=7$
$\Rightarrow a+b=c$
On squaring both sides, we get
$(a+b{)}^2=(-c{)}^2$
$\Rightarrow |a+b{|}^2=|c{|}^2$
$\Rightarrow |a{|}^2+|b{|}^2+2a\cdot b=|c{|}^2$
$(\because \theta$ be the angle between a and b)
$\Rightarrow (5{)}^2+(3{)}^2+2|a||b|\cos \theta =(7{)}^2$
$\Rightarrow 25+9+2\cdot 5\cdot 3\cos \theta =49$
$\Rightarrow 30\cos \theta =15$
$\Rightarrow \cos \theta =\frac{1}{2}=\cos {60}^{\circ }$
$\Rightarrow \theta =\frac{\pi }{3}$