Question

Assertion :If $\overrightarrow{r}\times \overrightarrow{a}=\overrightarrow{b},\overrightarrow{r}.\overrightarrow{a}=1,$ then $\overrightarrow{r}=\overrightarrow{a}+\overrightarrow{a}\times \overrightarrow{b}.$ Reason: The solution of the $\overrightarrow{r}\times \overrightarrow{a}=\overrightarrow{b}$ is $\overrightarrow{r}=\overrightarrow{a}\times \overrightarrow{b}$.

• 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 C Assertion is true but Reason is false

If $\overline{r}\times \overline{a}=\overline{b},\overline{r}\cdot \overline{a}=1,then\overline{r}=\overline{a}+\overline{a}\times \overline{b}$
The solution of the equation is
$\overline{r}\times \overline{a}=\overline{b}$
$i.e.\overline{a}(\overline{r}\times \overline{a})=\overline{a}\cdot \overline{b}$
$i.e.\overline{a}\cdot \overline{b}=0$
$\overline{r}=\alpha \overline{a}+\frac{\overline{a}\times \overline{b}}{\overline{a}\cdot \overline{a}}\Rightarrow \overline{r}\cdot \overline{a}=\alpha \overline{a}\cdot \overline{a}+\frac{\overline{a}\cdot (\overline{a}\times \overline{b})}{\overline{a}\cdot \overline{a}}$
$\Rightarrow 1=\alpha$
$\therefore \overline{r}=\overline{a}+\overline{a}\times \overline{b}$
Thus Assertion (A) is true & Reason (R) is false.