Q72 / 150 · MHT CET 2012 · Mathematics

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

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

Solution

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

Hence, the correct answer is option: 'B'.

Relevant Questions