Question

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ are two perpendicular unit vectors. If $\overrightarrow{c}$ is another unit vector equally inclined at angle q to the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ , then the set of exhaustive values of q in [0, 2p] is

• A. $\left( 0,\frac{\pi}{2} \right)$
• B. $\left\lbrack 0,\frac{\pi}{4} \right\rbrack$
• C. $\left( \frac{\pi}{2},\frac{3\pi}{4} \right)$
• D. $\left\lbrack \frac{\pi}{4},\frac{3\pi}{4} \right\rbrack$

Answer 1

Answer: (D)

$\left\lbrack \frac{\pi}{4},\frac{3\pi}{4} \right\rbrack$

Answer 2

Let $\overrightarrow{c}$= l $\overrightarrow{a}$+ m$\overrightarrow{b}$+ n($\overrightarrow{a}$×$\overrightarrow{b}$), then

$\overrightarrow{a}$.$\overrightarrow{c}$= l ̃ l = cos q

Similarly m = cos q

also, |$\overrightarrow{c}$|² = l² + m² + n² |$\overrightarrow{a}$×$\overrightarrow{b}$|²

̃ 1 = 2l² + n² [$|\overrightarrow{a}|^{2}|\overrightarrow{b}|^{2} - (\overrightarrow{a}.\overrightarrow{b})^{2}$] = 2l² + n² [Q l = m]

\ v² = 1 –2l² = 1 – 2 cos² q = –cos2q

Now n² ³ 0 ̃ cos 2q £ 0 ̃ $\frac{\pi}{4}$£ q £ $\frac{3\pi}{4}$