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Question
Find the cosine of the angle between the vectors \[4 \hat{i} - 3 \hat{j} + 3 \hat{k} \text{ and } 2 \hat{i} - \hat{j} - \hat{k} .\]
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Solution
\[\text{ Let }, \vec{a} = 4 \hat{i} - 3 \hat{j} + 3 \hat{k} \]
\[\text{ and } \vec{b} = 2 \hat{i} - \hat{j} - \hat{k} \]
\[\text{ Let }\theta\text{ be the angle between } \vec{a} \text{ and } \vec{b} . \]
\[\left| \vec{a} \right| = \sqrt{\left( 4 \right)^2 + \left( - 3 \right)^2 + \left( 3 \right)^2} = \sqrt{34}\]
\[\left| \vec{b} \right| = \sqrt{\left( 2 \right)^2 + \left( - 1 \right)^2 + \left( - 1 \right)^2} = \sqrt{6}\]
\[ \therefore \vec{a} . \vec{b} = 8 + 3 - 3 = 8\]
\[\cos \theta = \frac{\vec{a} . \vec{b}}{\left| \vec{a} \right| \left| \vec{b} \right|} = \frac{8}{\sqrt{34}\sqrt{6}} = \frac{8}{2\sqrt{51}} = \frac{4}{\sqrt{51}}\]
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