Question

Assertion :Let $\left|\overrightarrow{a}\right|=3,\left|\overrightarrow{b}\right|=5,\left|\overrightarrow{c}\right|=7$ such that $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0$ then angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{2\pi }{3}$ Reason: ${(\overrightarrow{x}+\overrightarrow{y})}^2={\left|\overrightarrow{x}\right|}^2+{\left|\overrightarrow{y}\right|}^2+2(\overrightarrow{x}\cdot \overrightarrow{y})$.

• A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
• B. Both Assertion & Reason are individually true but Reason is not the correct explanation of Assertion
• C. Assertion is true but Reason is false
• D. Assertion is false but Reason is true

Answer 1

The correct option is D Assertion is false but Reason is true

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

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

taking square on both sides
$\Rightarrow {(\overrightarrow{a}+\overrightarrow{b})}^2=-{\overline{c}}^2$

Putting values
$\Rightarrow {\left|\overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2+2\overrightarrow{a}\cdot \overrightarrow{b}={\left|\overrightarrow{c}\right|}^2$
$\Rightarrow 9+25+2\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta =49$
$\Rightarrow 2\cdot 3\cdot 5\cos \theta =15$
$\Rightarrow \cos \theta =\frac{1}{2}$

So, $\theta =\frac{\pi }{3}\ne \frac{2\pi }{3}$
Assertion $\left(A\right)$ is false and Reason $\left(R\right)$ is true