If a unit vector a makes angles 3 with i, 4 with j, and (0,) with k, a value of is:

Question

If a unit vector $\vec a$ makes angles $\dfrac\pi3$ with $\hat i$, $\dfrac\pi4$ with $\hat j$, and $	heta\in(0,\pi)$ with $\hat k$, a value of $	heta$ is:

Accepted Answer

Direction cosines satisfy $\cos^2\dfrac\pi3+\cos^2\dfrac\pi4+\cos^2\theta=1$:
$$\frac14+\frac12+\cos^2\theta=1 \implies \cos^2\theta=\frac14 \implies \cos\theta=\pm\frac12 \implies \theta=\frac\pi3\text{ or }\frac{2\pi}3$$

Since $\theta=\pi/3$ is not listed, the valid answer is $\theta=\dfrac{2\pi}3$.

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