Question

Assertion :If $\overline{a},\overline{b},\overline{c}$ are unit vectors such that $\overline{a}+\overline{b}+\overline{c}=0$, then $\overline{a}\cdot \overline{b}+\overline{b}\cdot \overline{c}+\overline{c}\cdot \overline{a}=\frac{-3}{2}.$ Reason: ${(\overline{x}+\overline{y})}^2={\overline{x}}^2+{\overline{y}}^2+2\overline{x}\cdot \overline{y}$

• 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 A Both Assertion & Reason are individually true & Reason is correct explanation of Assertion

We have

$\left|\overline{a}\right|=\left|\overline{b}\right|=\left|\overline{c}\right|=1$ and $\overline{a}+\overline{b}+\overline{c}=0$

$\Rightarrow {(\overline{a}+\overline{b}+\overline{c})}^2=0\Rightarrow 3+2(\overline{a}\cdot \overline{b}+\overline{b}\cdot \overline{c}+\overline{c}\cdot \overline{a})=0$

$\Rightarrow \overline{a}\cdot \overline{b}+\overline{b}\cdot \overline{c}+\overline{c}\cdot \overline{a}=\frac{-3}{2}$

$\Rightarrow$ Assertion $\left(A\right)$ is correct and obtained with the help of given Reason $\left(R\right)$ which means Reason $\left(R\right)$ is also correct with correct explanation of Assertion $\left(A\right)$