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Question

# Write a vector of magnitude 12 units which makes 45° angle with X-axis, 60° angle with Y-axis and an obtuse angle with Z-axis.

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Solution

## Suppose a vector $\stackrel{\to }{r}$ makes an angle 45$°$ with $ΟΧ$, 60$°$ with $ΟΥ$ and having magnitude 12 units. $l=\mathrm{cos}45°=\frac{1}{\sqrt{2}}\mathrm{and}m=\mathrm{cos}60°=\frac{1}{2}$ $\mathrm{Now},{l}^{2}+{m}^{2}+{n}^{2}=1\phantom{\rule{0ex}{0ex}}⇒\frac{1}{2}+\frac{1}{4}+{n}^{2}=1\phantom{\rule{0ex}{0ex}}⇒{n}^{2}=\frac{1}{4}\phantom{\rule{0ex}{0ex}}⇒n=-\frac{1}{2}\left[\because \mathrm{The}\mathrm{angle}\mathrm{with}\mathrm{the}z-\mathrm{axis}\mathrm{is}\mathrm{obtuse}\right]$ Therefore, $\stackrel{\to }{r}=\left|\stackrel{\to }{r}\right|\left(l\stackrel{^}{i}+m\stackrel{^}{j}+n\stackrel{^}{k}\right)\phantom{\rule{0ex}{0ex}}=12\left(\frac{1}{\sqrt{2}}\stackrel{^}{i}+\frac{1}{2}\stackrel{^}{j}-\frac{1}{2}\stackrel{^}{k}\right)\phantom{\rule{0ex}{0ex}}=6\left(\sqrt{2}\stackrel{^}{i}+\stackrel{^}{j}-\stackrel{^}{k}\right)$

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