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Question

# A vector of magnitude 6, making angle $\frac{\mathrm{\pi }}{4}\mathrm{with}x-\mathrm{axis},\frac{\mathrm{\pi }}{3}$ with y-axis and an acute angle with z-axis is ____________.

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Solution

## Given: A vector of magnitude 6, making angle $\frac{\mathrm{\pi }}{4}\mathrm{with}x-\mathrm{axis},\frac{\mathrm{\pi }}{3}$ with y-axis and an acute angle with z-axis Let a vector $\stackrel{\to }{a}$ makes an angle $\frac{\mathrm{\pi }}{4}\mathrm{with}x-\mathrm{axis},\frac{\mathrm{\pi }}{3}$ with y-axis and an acute angle with z-axis and is of magnitude 6. $\mathrm{Therefore},\phantom{\rule{0ex}{0ex}}l=\mathrm{cos}\frac{\mathrm{\pi }}{4}=\frac{1}{\sqrt{2}}\phantom{\rule{0ex}{0ex}}m=\mathrm{cos}\frac{\mathrm{\pi }}{3}=\frac{1}{2}\phantom{\rule{0ex}{0ex}}n=\mathrm{cos}\theta \phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{We}\mathrm{know},\phantom{\rule{0ex}{0ex}}{l}^{2}+{m}^{2}+{n}^{2}=1\phantom{\rule{0ex}{0ex}}⇒{\left(\frac{1}{\sqrt{2}}\right)}^{2}+{\left(\frac{1}{2}\right)}^{2}+{\left(\mathrm{cos}\theta \right)}^{2}=1\phantom{\rule{0ex}{0ex}}⇒\frac{1}{2}+\frac{1}{4}+{\left(\mathrm{cos}\theta \right)}^{2}=1\phantom{\rule{0ex}{0ex}}⇒{\left(\mathrm{cos}\theta \right)}^{2}=1-\frac{1}{2}-\frac{1}{4}\phantom{\rule{0ex}{0ex}}⇒{\left(\mathrm{cos}\theta \right)}^{2}=\frac{4-2-1}{4}\phantom{\rule{0ex}{0ex}}⇒{\left(\mathrm{cos}\theta \right)}^{2}=\frac{1}{4}\phantom{\rule{0ex}{0ex}}⇒\mathrm{cos}\theta =\frac{1}{2}\left(\because \theta \mathrm{is}\mathrm{acute}\right)\phantom{\rule{0ex}{0ex}}⇒n=\frac{1}{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Thus,\phantom{\rule{0ex}{0ex}}\stackrel{\to }{a}=\left|\stackrel{\to }{a}\right|\left(l\stackrel{^}{i}+m\stackrel{^}{j}+n\stackrel{^}{k}\right)\phantom{\rule{0ex}{0ex}}=6\left(\frac{1}{\sqrt{2}}\stackrel{^}{i}+\frac{1}{2}\stackrel{^}{j}+\frac{1}{2}\stackrel{^}{k}\right)\phantom{\rule{0ex}{0ex}}=3\sqrt{2}\stackrel{^}{i}+3\stackrel{^}{j}+3\stackrel{^}{k}$ Hence, a vector of magnitude 6, making angle $\frac{\mathrm{\pi }}{4}\mathrm{with}x-\mathrm{axis},\frac{\mathrm{\pi }}{3}$ with y−axis and an acute angle with z−axis is $\overline{)3\sqrt{2}\stackrel{^}{i}+3\stackrel{^}{j}+3\stackrel{^}{k}}.$

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