Vector Algebra
Dot Product and Projection
Grade None
Question:
<p>[JEE Main 2021] If a unit vector \(\hat{a}\) makes angles \(\dfrac\pi3\) with \(\hat{i}\), \(\dfrac\pi4\) with \(\hat{j}\) and an acute angle \(\theta\) with \(\hat{k}\), then \(\theta\) equals</p>
<li>\(\dfrac{\pi}{4}\)</li>
<li>\(\dfrac{\pi}{3}\)</li>
<li>\(\dfrac{\pi}{6}\)</li>
<li>\(\dfrac{5\pi}{12}\)</li>
Step-by-Step Solution
Key Concept: Direction cosines: l=cos(\pi/3)=½, m=cos(\pi/4)=1/\sqrt{2.} Use l^2+m^2+n^2=1 to find n=cos\theta.
$l=\cos\dfrac{\pi}{3}=\dfrac12,\;m=\cos\dfrac{\pi}{4}=\dfrac{1}{\sqrt2}$.
$l^2+m^2+n^2=1\Rightarrow\dfrac14+\dfrac12+n^2=1\Rightarrow n^2=\dfrac14\Rightarrow n=\pm\dfrac12$.
Since $\theta$ is acute: $n=\cos\theta=\dfrac12>0\Rightarrow\theta=\dfrac{\pi}{3}$. Answer: (B)
Correct Answer: B