MHT CET · Maths · Vector Algebra
A unit vector \(\hat{a}\) makes angles \(\frac{\pi}{3}\) with \(\hat{i}, \frac{\pi}{4}\) with \(\hat{j}\) and \(\theta \in(\theta, \pi)\) with \(\widehat{k}\), then a value of \(\theta\) is
- A \(\frac{2 \pi}{3}\)
- B \(\frac{5 \pi}{12}\)
- C \(\frac{5 \pi}{6}\)
- D \(\frac{\pi}{4}\)
Answer & Solution
Correct Answer
(A) \(\frac{2 \pi}{3}\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \alpha=\frac{\pi}{3}, \beta=\frac{\pi}{4} \text { and } \gamma=0 \\ & \because \cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1 \\ & \Rightarrow \cos ^2 \frac{\pi}{3}+\cos ^2 \frac{\pi}{4}+\cos ^2 \theta=1 \\ & \Rightarrow \frac{1}{4}+\frac{1}{2}+\cos ^2 \theta=1\end{aligned}\)
\(\Rightarrow \cos ^2 \theta=\frac{1}{4}\)
\(\Rightarrow \cos \theta= \pm \frac{1}{2}\)
\(\Rightarrow \theta=\frac{\pi}{3}\) or \(\frac{2 \pi}{3}\) as \(\theta \in(0, \pi)\)
\(\Rightarrow \cos ^2 \theta=\frac{1}{4}\)
\(\Rightarrow \cos \theta= \pm \frac{1}{2}\)
\(\Rightarrow \theta=\frac{\pi}{3}\) or \(\frac{2 \pi}{3}\) as \(\theta \in(0, \pi)\)
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