Question

Assertion :Let $\overline{a},\overline{b},\overline{r}$ be the vectors such that $\overline{r}+\overline{r}\times \overline{a}=\overline{b}$ then ${\left|\overline{r}\right|}^2=\frac{{(\overline{a}.\overline{b})}^2+{\left|\overline{b}\right|}^2}{1+{\left|\overline{a}\right|}^2}$. Reason: $\overline{r}=\frac{(\overline{a}.\overline{b})\overline{a}+\overline{b}+\overline{a}\times \overline{b}}{1+{\left|\overline{a}\right|}^2}.$

• 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 (proper) 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

$\overline{r}+\overline{r}\times \overline{a}=\overline{b}$

Taking dot product on both sides
$\Rightarrow \overline{r}\cdot \overline{a}+(\overline{r}\times \overline{a})\cdot \overline{a}=\overline{a}\cdot \overline{b}$
$\Rightarrow \overline{r}\cdot \overline{a}=\overline{a}\cdot \overline{b}$ .......( * )

and

$\Rightarrow {|\overline{r}+\overline{r}\times \overline{a}|}^2={\left|\overline{b}\right|}^2$
$\Rightarrow \overline{r}\cdot \overline{r}+(\overline{r}\times \overline{a})\cdot (\overline{r}\times \overline{a})=\overline{b}\cdot \overline{b}$
$\Rightarrow \overline{r}\cdot \overline{r}+{|\overline{r}\times \overline{a}|}^2=\overline{b}\cdot \overline{b}$
$\Rightarrow \overline{r}\cdot \overline{r}+{\overline{r}}^2{\overline{a}}^2{\sin }^2\theta =\overline{b}\cdot \overline{b}$
$\Rightarrow \overline{r}\cdot \overline{r}+{\left|\overline{r}\right|}^2{\left|\overline{a}\right|}^2-{\left|\overline{r}\right|}^2{\left|\overline{a}\right|}^2{\cos }^2\theta =\overline{b}\cdot \overline{b}$
$\Rightarrow \overline{r}\cdot \overline{r}+(\overline{r}\cdot \overline{r})(\overline{a}\cdot \overline{a})-{(\overline{r}\cdot \overline{a})}^2=\overline{b}\cdot \overline{b}$
$\Rightarrow \overline{r}\cdot \overline{r}=\frac{{(\overline{a}\cdot \overline{b})}^2+\overline{b}\cdot \overline{b}}{1+\overline{a}\cdot \overline{a}}=\frac{{(\overline{a}\cdot \overline{b})}^2+\overline{b}\cdot \overline{b}}{1+{\left|\overline{a}\right|}^2}$
Hence option $A$ is correct