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Question

# A vector $\stackrel{\to }{r}$ is inclined at equal acute angles to x-axis, y-axis and z-axis. If $\left|\stackrel{\to }{r}\right|$ = 6 units, find $\stackrel{\to }{r}$.

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Solution

## Suppose, vector $\stackrel{\to }{r}$ makes an angle $\alpha$ with each of the axis $OX,OY$ and $OZ$. Then, its direction cosines are $l=\mathrm{cos}\alpha ,m=\mathrm{cos}\alpha$ and $n=\mathrm{cos}\alpha$ i.e. $l=m=n.$ $\mathrm{Now},{l}^{2}+{m}^{2}+{n}^{2}=1\phantom{\rule{0ex}{0ex}}⇒{l}^{2}+{l}^{2}+{l}^{2}=1\phantom{\rule{0ex}{0ex}}⇒3{l}^{2}=1\phantom{\rule{0ex}{0ex}}⇒{l}^{2}=\frac{1}{3}\phantom{\rule{0ex}{0ex}}⇒l=±\frac{1}{\sqrt{3}}\phantom{\rule{0ex}{0ex}}\mathrm{Since},\stackrel{\to }{r}\mathrm{makes}\mathrm{acute}\mathrm{angle}\mathrm{with}\mathrm{the}\mathrm{axis}.\phantom{\rule{0ex}{0ex}}\mathrm{Hence},\mathrm{we}\mathrm{take}\mathrm{only}\mathrm{positive}\mathrm{value}.\phantom{\rule{0ex}{0ex}}$ Therefore, $\stackrel{\to }{r}=\left|\stackrel{\to }{r}\right|\left(l\stackrel{^}{i}+m\stackrel{^}{j}+n\stackrel{^}{k}\right)\phantom{\rule{0ex}{0ex}}\stackrel{\to }{r}=6\left(\frac{1}{\sqrt{3}}\stackrel{^}{i}+\frac{1}{\sqrt{3}}\stackrel{^}{j}+\frac{1}{\sqrt{3}}\stackrel{^}{k}\right)\phantom{\rule{0ex}{0ex}}=2\sqrt{3}\left(\stackrel{^}{i}+\stackrel{^}{j}+\stackrel{^}{k}\right)$

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