Question

Let $\overset{\rightarrow}{r}$ be a unit vector satisfying $\overset{\rightarrow}{r}$× $\overset{\rightarrow}{a}$ = $\overset{\rightarrow}{b}$, where

|$\overset{\rightarrow}{a}$| = $\sqrt{3}$ and |$\overset{\rightarrow}{b}$| = $\sqrt{2}$, Then :

• A. $\overset{\rightarrow}{r}$ = $\frac{2}{3}$ ($\overset{\rightarrow}{a}$+$\overset{\rightarrow}{a}$×$\overset{\rightarrow}{b}$)
• B. $\overset{\rightarrow}{r}$= $\frac{1}{3}$ ($\overset{\rightarrow}{a}$× $\overset{\rightarrow}{b}$± $\overset{\rightarrow}{a}$ )
• C. $\overset{\rightarrow}{r}$ = ($\overset{\rightarrow}{a}$ × $\overset{\rightarrow}{b}$± a )
• D. $\overset{\rightarrow}{r}$ = ($\overset{\rightarrow}{b}$±$\overset{\rightarrow}{a}$×$\overset{\rightarrow}{b}$)

Answer 1

Answer: (B)

$\overset{\rightarrow}{r}$= $\frac{1}{3}$ ($\overset{\rightarrow}{a}$× $\overset{\rightarrow}{b}$± $\overset{\rightarrow}{a}$ )

Answer 2

$\overset{\rightarrow}{r}$ × $\overset{\rightarrow}{a}$ = $\overset{\rightarrow}{b}$ ̃ $\overset{\rightarrow}{a}$× ($\overset{\rightarrow}{r}$×$\overset{\rightarrow}{a}$) = $\overset{\rightarrow}{a}$ × $\overset{\rightarrow}{b}$

($\overset{\rightarrow}{a}$.$\overset{\rightarrow}{a}$)$\overset{\rightarrow}{r}$ – ($\overset{\rightarrow}{a}$.$\overset{\rightarrow}{r}$)$\overset{\rightarrow}{a}$ = $\overset{\rightarrow}{a}$× $\overset{\rightarrow}{b}$

3$\overset{\rightarrow}{r}$ = ($\overset{\rightarrow}{a}$.$\overset{\rightarrow}{r}$)$\overset{\rightarrow}{a}$ + ($\overset{\rightarrow}{a}$× $\overset{\rightarrow}{b}$) …(i)

Now, |$\overset{\rightarrow}{r}$ × $\overset{\rightarrow}{a}$|² = |$\overset{\rightarrow}{b}$|² = 2

or, 2 = ($\overset{\rightarrow}{r}$×$\overset{\rightarrow}{a}$) . ($\overset{\rightarrow}{r}$×$\overset{\rightarrow}{a}$)

= $\left| \begin{matrix} \overset{\rightarrow}{r}.\overset{\rightarrow}{r} & \overset{\rightarrow}{a}.\overset{\rightarrow}{r} \\ \overset{\rightarrow}{a}.\overset{\rightarrow}{r} & \overset{\rightarrow}{a}.\overset{\rightarrow}{a} \end{matrix} \right|$

̃ $\overset{\rightarrow}{r}$.$\overset{\rightarrow}{a}$ = ± 1. …(ii)

\ $\overset{\rightarrow}{r}$ = $\frac{1}{3}$ ($\overset{\rightarrow}{a}$ × $\overset{\rightarrow}{b}$ ± $\overset{\rightarrow}{a}$)

from (i) & (ii)