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Question

# Find a vector $\stackrel{\to }{r}$ of magnitude $3\sqrt{2}$ units which makes an angle of $\frac{\mathrm{\pi }}{4}$ and $\frac{\mathrm{\pi }}{2}$ with y and z-axes respectively. [NCERT EXEMPLAR]

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Solution

## Suppose vector $\stackrel{\to }{r}$ makes an angle α with the x-axis. Let l, m, n be the direction cosines of $\stackrel{\to }{r}$. Then, $l=\mathrm{cos}\alpha ,m=\mathrm{cos}\frac{\mathrm{\pi }}{4}=\frac{1}{\sqrt{2}},n=\mathrm{cos}\frac{\mathrm{\pi }}{2}=0$ Now, ${l}^{2}+{m}^{2}+{n}^{2}=1\phantom{\rule{0ex}{0ex}}⇒{\mathrm{cos}}^{2}\alpha +\frac{1}{2}+0=1\phantom{\rule{0ex}{0ex}}⇒{\mathrm{cos}}^{2}\alpha =1-\frac{1}{2}=\frac{1}{2}\phantom{\rule{0ex}{0ex}}⇒\mathrm{cos}\alpha =±\frac{1}{\sqrt{2}}$ We know that $\stackrel{\to }{r}=\left|\stackrel{\to }{r}\right|\left(l\stackrel{^}{i}+m\stackrel{^}{j}+n\stackrel{^}{k}\right)\phantom{\rule{0ex}{0ex}}\therefore \stackrel{\to }{r}=3\sqrt{2}\left(±\frac{1}{\sqrt{2}}\stackrel{^}{i}+\frac{1}{\sqrt{2}}j+0\stackrel{^}{k}\right)\left(\left|\stackrel{\to }{r}\right|=3\sqrt{2}\right)\phantom{\rule{0ex}{0ex}}⇒\stackrel{\to }{r}=±3\stackrel{^}{i}+3\stackrel{^}{j}$

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