Vectors 1 Question 2
2. If a unit vector a makes angles $\frac{\pi}{3}$ with $\hat{\mathbf{i}}, \frac{\pi}{4}$ with $\hat{\mathbf{j}}$ and $\theta \in(0, \pi)$ with $\hat{\mathbf{k}}$, then a value of $\theta$ is
(2019 Main, 9 April II)
(a) $\frac{5 \pi}{6}$
(b) $\frac{\pi}{4}$
(c) $\frac{5 \pi}{12}$
(d) $\frac{2 \pi}{3}$
Show Answer
Answer:
Correct Answer: 2. (d)
Solution:
- Given unit vector a makes an angle $\frac{\pi}{3}$ with $\hat{\mathbf{i}}, \frac{\pi}{4}$ with $\hat{\mathbf{j}}$ and $\theta \in(0, \pi)$ with $\hat{\mathbf{k}}$.
Now, we know that $\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$, where $\alpha, \beta, \gamma$ are angles made by the vectors
with respectively $\hat{\mathbf{i}}, \hat{\mathbf{j}}$ and $\hat{\mathbf{k}}$.
$\therefore \cos ^{2}\left(\frac{\pi}{3}\right)+\cos ^{2}\left(\frac{\pi}{4}\right)+\cos ^{2} \theta=1$
$\Rightarrow \frac{1}{4}+\frac{1}{2}+\cos ^{2} \theta=1 \Rightarrow \cos ^{2} \theta=\frac{1}{4} \Rightarrow \cos \theta= \pm \frac{1}{2}$
$\Rightarrow \cos \theta=\cos \left(\frac{\pi}{3}\right)$ or $\cos \left(\frac{2 \pi}{3}\right) \Rightarrow \theta=\frac{\pi}{3}$ or $\frac{2 \pi}{3}$
So, $\theta$ is $\frac{2 \pi}{3}$, according to options.
