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If a unit vector makes angles with with and an acute angle with then find and hence, the components of .

Q3   If a unit vector \vec a makes angles \frac{\pi }{3}with\hat i , \frac{\pi }{4}   with \hat j and an acute angle \theta \: \: with \hat k  then find \theta \: \:  and hence, the components of \vec a.

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Given in the question,

angle between \vec a and \hat i :

 \alpha =\frac{\pi}{3}

angle between \vec a and \hat j

\beta =\frac{\pi}{4}

angle with \vec a and \hat k:

\gamma =\theta

Now, As we know,

cos^2\alpha+cos^2\beta+cos^2\gamma=1

cos^2\frac{\pi}{3}+cos^2\frac{\pi }{4}+cos^2\theta=1

\left ( \frac{1}{2} \right )^2+\left ( \frac{1}{\sqrt{2}} \right )^2+cos^2\theta=1

cos^2\theta=\frac{1}{4}

cos\theta=\frac{1}{2}

\theta=\frac{\pi}{3}

Now components of \vec a are:

\left ( cos\frac{\pi}{3},cos\frac{\pi}{2},cos\frac{\pi}{3} \right )=\left ( \frac{1}{2},\frac{1}{\sqrt{2}},\frac{1}{2} \right ).

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