Search any question & find its solution
Question:
Answered & Verified by Expert
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
Options:
then angle between $\mathbf{a}$ and $\mathbf{b}$ is
Solution:
1289 Upvotes
Verified Answer
The correct answer is:
$\frac{\pi}{3}$
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 $\mathbf{b}$ )
$\Rightarrow(5)^{2}+(3)^{2}+2|\mathbf{a}||\mathbf{b}| \cos \theta=(7)^{2}$
$\Rightarrow \quad 25+9+2 \cdot 5 \cdot 3 \cdot \cos \theta=49$
$\Rightarrow \quad 30 \cos \theta=15$
$\Rightarrow \quad \cos \theta=\frac{1}{2}=\cos 60^{\circ}$
$\Rightarrow \quad \theta=\frac{\pi}{3}$
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 $\mathbf{b}$ )
$\Rightarrow(5)^{2}+(3)^{2}+2|\mathbf{a}||\mathbf{b}| \cos \theta=(7)^{2}$
$\Rightarrow \quad 25+9+2 \cdot 5 \cdot 3 \cdot \cos \theta=49$
$\Rightarrow \quad 30 \cos \theta=15$
$\Rightarrow \quad \cos \theta=\frac{1}{2}=\cos 60^{\circ}$
$\Rightarrow \quad \theta=\frac{\pi}{3}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.