Question

If three vectors $a,b,c$ satisfy $a+b+c=0$ and $|a|=3,|b|=5,|c|=7,$ then the angle between $a$ and $b$ is :

• A. ${30}^o$
• B. ${45}^o$
• C. ${60}^o$
• D. ${90}^o$

Answer 1

The correct option is C ${60}^o$

Given, $|a|=3,|b|=5,|c|=7$ and

$\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0$

=> $\overrightarrow{a}+\overrightarrow{b}=(-\overrightarrow{c})$

On squaring both side, we get

${(\overrightarrow{a}+\overrightarrow{b})}^2={(-\overrightarrow{c})}^2$

=> ${|\overrightarrow{a}|}^2+{|\overrightarrow{b}|}^2+2(\overrightarrow{a}.\overrightarrow{b})={|\overrightarrow{c}|}^2$

Let the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\theta$, then

${|\overrightarrow{a}|}^2+{|\overrightarrow{b}|}^2+2|\overrightarrow{a}||\overrightarrow{b}|\cos \theta ={|\overrightarrow{c}|}^2$

=> $3^2+5^2+2\times 3\times 5\times \cos \theta =7^2$

=> $\cos \theta =\frac{1}{2}$

So, the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\theta$ $={60}^o$