A unit vector r makes angles 3 and 2 with j and k respectively an… - Vidyadip

Question

A unit vector $\vec{\text{r}}$ makes angles $\frac{\pi}3\text{ and }\frac{\pi}2$ with $\hat{\text{j}}\text{ and } \hat{\text{k}}$ respectively and an acute angle $\theta$ with $\hat{\text{i}}$. Find $\theta$.

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Answer

A unit vector makes an angle $\frac{\pi}3\text{ and }\frac{\pi}2$ with $\hat{\text{j}}\text{ and } \hat{\text{k}}$Let l, m, n be its direction cosines
$\therefore\ \text{l}=\cos\theta,\ \text{m}=\cos\big(\frac{\pi}3\big)=\frac{1}2,\ \text{n}=\cos\big(\frac{\pi}2\big)=0$
Now,
$\text{l}^2+\text{m}^2+\text{n}^2=1$
$\Rightarrow\ \text{l}^2+\frac{1}4+0=1$
$\Rightarrow\ \text{l}^2=1-\frac{1}4=\frac{3}4$
$\Rightarrow\ \text{l}=\pm\frac{\sqrt3}2$
$\therefore\ \vec{\text{r}}$ makes an acute angle 30º, 150º with $\hat{\text{i}}$
Since, angle $\theta$ is acute.
$\therefore\ \theta=30^{\circ}$

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