Question

Attempt any three part of the following :
If $|\overrightarrow{a}|=3,|\overrightarrow{b}|=5and|\overrightarrow{c}|=7$ and $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0$ then prove that angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\pi }{3}$

Answer 1

To Prove that angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\pi }{3}$

Given , $\left|\overrightarrow{a}\right|=3,\left|\overrightarrow{b}\right|=5$ and $\left|\overrightarrow{c}\right|=7$ and $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0$

This can be written as

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

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

On Squaring both sides, we get

${\left|\overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2+2\overrightarrow{a}.\overrightarrow{b}={\left|\overrightarrow{c}\right|}^2$

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

$(3{)}^2+(5{)}^2+2.3.5cos\theta =(7{)}^2$

$9+25+30cos\theta =49$

$34+30cos\theta =49$

$30cos\theta =15$

$cos\theta =\frac{15}{30}$

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

$\theta =\frac{\pi }{3}$

Hence , angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\pi }{3}$