Question

Let $\overrightarrow{r}\times \overrightarrow{a}=\overrightarrow{b}\times \overrightarrow{a}$ and $\overrightarrow{r}\cdot \overrightarrow{c}=0$ where $\overrightarrow{a}\cdot \overrightarrow{c}\ne 0$ then find the value of $(\overrightarrow{a}\cdot \overrightarrow{c})(\overrightarrow{r}\times \overrightarrow{b})+(\overrightarrow{a}\times \overrightarrow{r})$

• A. $c$
• B. $(\overrightarrow{a}\cdot \overrightarrow{b})c$
• C. $(\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{c}$
• D. $0$

Answer 1

Answer: (D)

$0$

Given $(\overrightarrow{r}-\overrightarrow{b})\times \overrightarrow{a}=0$
$\Rightarrow \overrightarrow{r}-\overrightarrow{b}$ and $\overrightarrow{a}$ are parallel.
$\therefore \overrightarrow{r}-\overrightarrow{b}=t\overrightarrow{a}$ or $\therefore \overrightarrow{r}=\overrightarrow{b}+t\overrightarrow{a}..................\left(1\right)$ or
$\therefore \overrightarrow{r}\cdot \overrightarrow{c}=(\overrightarrow{b}\cdot \overrightarrow{c})+t(\overrightarrow{a}\cdot \overrightarrow{c})$
or $\therefore 0=(\overrightarrow{b}\cdot \overrightarrow{c})+t(\overrightarrow{a}\cdot \overrightarrow{c})\therefore t=-\frac{(\overrightarrow{b}\cdot \overrightarrow{c})}{(\overrightarrow{a}\cdot \overrightarrow{c})}...............\left(2\right)$

Hence from $\left(1\right)$ and $\left(2\right),$ putting for $t$
$\overrightarrow{r}=\overrightarrow{b}-\frac{\overrightarrow{b}\cdot \overrightarrow{c}}{\overrightarrow{a}\cdot \overrightarrow{c}}\overrightarrow{a}$
$\therefore (\overrightarrow{r}\times \overrightarrow{b})=(\overrightarrow{b}\times \overrightarrow{b})-\frac{\overrightarrow{b}\cdot \overrightarrow{c}}{\overrightarrow{a}\cdot \overrightarrow{c}}(\overrightarrow{a}\times \overrightarrow{b})$
$\therefore (\overrightarrow{a}\cdot \overrightarrow{c})(\overrightarrow{r}\times \overrightarrow{b})+(\overrightarrow{b}\cdot \overrightarrow{c})(\overrightarrow{a}\times \overrightarrow{b})=0$