A unit vector hat{a} makes angles frac{pi}{3} | vedclass

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Question

(A) Let the direction angles of the unit vector $\hat{a}$ be $\alpha, \beta, 	ext{ and } \gamma$.Given $\alpha = \frac{\pi}{3}$,$\beta = \frac{\pi}{4

Vedclass · Accepted Answer

$\frac{2 \pi}{3}$

Vedclass · Answer

(A) Let the direction angles of the unit vector $\hat{a}$ be $\alpha, \beta, 	ext{ and } \gamma$.Given $\alpha = \frac{\pi}{3}$,$\beta = \frac{\pi}{4}$,and $\gamma = 	heta$.We know that for any unit vector,the sum of the squares of the direction cosines is $1$:$\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$Substituting the given values:$\cos^2 \left(\frac{\pi}{3}ight) + \cos^2 \left(\frac{\pi}{4}ight) + \cos^2 	heta = 1$$\left(\frac{1}{2}ight)^2 + \left(\frac{1}{\sqrt{2}}ight)^2 + \cos^2 	heta = 1$$\frac{1}{4} + \frac{1}{2} + \cos^2 	heta = 1$$\frac{3}{4} + \cos^2 	heta = 1$$\cos^2 	heta = 1 - \frac{3}{4} = \frac{1}{4}$$\cos 	heta = \pm \frac{1}{2}$Since $	heta \in (0, \pi)$,we have two possible values:If $\cos 	heta = \frac{1}{2}$,then $	heta = \frac{\pi}{3}$.If $\cos 	heta = -\frac{1}{2}$,then $	heta = \frac{2\pi}{3}$.Comparing with the given options,$\frac{2\pi}{3}$ is the correct value.

$A$ unit vector $\hat{a}$ makes angles $\frac{\pi}{3}$ with $\hat{i}$,$\frac{\pi}{4}$ with $\hat{j}$,and $\theta \in (0, \pi)$ with $\hat{k}$. Then a value of $\theta$ is:

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