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Question
A unit vector \[\vec{a}\] makes angles \[\frac{\pi}{4}\text{ and }\frac{\pi}{3}\] with \[\hat{i}\] and \[\hat{j}\] respectively and an acute angle θ with \[\hat{k}\] . Find the angle θ and components of \[\vec{a}\] .
Solution
\[\text{ Let }\vec{a} = a_1 \hat{i} {} + a_2 \hat{j} {+a}_3 \hat{k} , \text{ where } a_1 , a_2 \text{ and } a_3 \text{ are components of } \vec{a} .\]
\[ \Rightarrow {a_1}^2 + {a_2}^2 + {a_3}^2 = 1............... ( \text{ Because }\vec{a} \text{ is a unit vector }) . . . \left( 1 \right)\]
\[\text{ Now },\]
\[ \vec{a} . \text{i} = a_1 \]
\[ \Rightarrow \left| \vec{a} \right|\left| \hat{i} \right| \cos \frac{\pi}{4} = a_1 (\text{ Because the angle between } \vec{a} \text{ and } \hat{i} \text{ is }\frac{\pi}{4})\]
\[ \Rightarrow \left( 1 \right) \left( 1 \right) \frac{1}{\sqrt{2}} = a_1 ...........(\text{ Because } \vec{a} \text{ and } \hat{i} \text{ are unit vectors })\]
\[ \Rightarrow a_1 = \frac{1}{\sqrt{2}}\]
\[\text{ Again },\]
\[ \vec{a} . \hat{j} = a_2 \]
\[ \Rightarrow \left| \vec{a} \right|\left| \hat{i} \right| \cos \frac{\pi}{3} = a_2........... (\text{ Because the angle between } \vec{a} \text{ and } \hat{i} \text{ is }\frac{\pi}{3})\]
\[ \Rightarrow \left( 1 \right) \left( 1 \right) \frac{1}{2} = a_2 ...........(\text{ Because } \vec{a} \text{ and } \hat{i} \text{ are unit vectors })\]
\[ \Rightarrow a_2 = \frac{1}{2}\]
\[\text{ Now from } (1),\]
\[ \left( \frac{1}{\sqrt{2}} \right)^2 + \left( \frac{1}{2} \right)^2 + {a_3}^2 = 1\]
\[ \Rightarrow \frac{1}{2} + \frac{1}{4} + {a_3}^2 = 1\]
\[ \Rightarrow \frac{3}{4} + {a_3}^2 = 1\]
\[ \Rightarrow {a_3}^2 = \frac{1}{4}\]
\[ \Rightarrow a_3 = \frac{1}{2}\]
\[\text{ Now },\]
\[ \vec{a} . \hat{k} = a_3 \]
\[ \Rightarrow \left| \vec{a} \right|\left| \hat{k} \right| \cos \theta = \frac{1}{2}.......... (\text{ Because the angle between } \vec{a} \text{ and } \hat{k} \text{ is }\theta)\]
\[ \Rightarrow \left( 1 \right) \left( 1 \right) \cos \theta = \frac{1}{2}..........(\text{ Because } \vec{a} \text{ and } \hat{i} \hat{\text{ are unit }}vectors)\]
\[ \Rightarrow \theta = \cos^{- 1} \left( \frac{1}{2} \right) = \frac{\pi}{3}\]
\[\text{ And } \]
\[ \vec{a} =\frac{1}{\sqrt{2}}\hat{i} {} + \frac{1}{2} \hat{j} + \frac{1}{2} \hat{k} \]
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