Question

Find the cosine of the angle between the vectors \[4 \hat{i} - 3 \hat{j} + 3 \hat{k} \text{ and } 2 \hat{i} - \hat{j} - \hat{k} .\] 

Answer 1

\[\text{ Let }, \vec{a} = 4 \hat{i} - 3 \hat{j} + 3 \hat{k} \]

\[\text{ and } \vec{b} = 2 \hat{i} - \hat{j} - \hat{k} \]

\[\text{ Let }\theta\text{ be the angle between } \vec{a} \text{ and } \vec{b} . \]

\[\left| \vec{a} \right| = \sqrt{\left( 4 \right)^2 + \left( - 3 \right)^2 + \left( 3 \right)^2} = \sqrt{34}\]

\[\left| \vec{b} \right| = \sqrt{\left( 2 \right)^2 + \left( - 1 \right)^2 + \left( - 1 \right)^2} = \sqrt{6}\]

\[ \therefore \vec{a} . \vec{b} = 8 + 3 - 3 = 8\]

\[\cos \theta = \frac{\vec{a} . \vec{b}}{\left| \vec{a} \right| \left| \vec{b} \right|} = \frac{8}{\sqrt{34}\sqrt{6}} = \frac{8}{2\sqrt{51}} = \frac{4}{\sqrt{51}}\]