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Question
Find a unit vector perpendicular to both the vectors \[4 \hat{ i } - \hat{ j } + 3 \hat{ k } \text{ and } - 2 \hat{ i } + \hat{ j } - 2 \hat{ k } .\]
Solution
\[ \text{ Given } : \]
\[ \vec{a} = 4 \hat { i } - \hat{ j } + 3 \hat{ k } \]
\[ \vec{b} = - 2 \hat{ i } + \hat{ j } - 2 \hat{ k } \]
\[ \therefore \vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k} \\ 4 & - 1 & 3 \\ - 2 & 1 & - 2\end{vmatrix}\]
\[ = \left( 2 - 3 \right) \hat{ i } - \left( - 8 + 6 \right) \hat{ j } + \left( 4 - 2 \right) \hat{ k } \]
\[ = - \hat{ i } + 2 \hat{ j } + 2 \hat{ k } \]
\[ \Rightarrow \left| \vec{a} \times \vec{b} \right| = \sqrt{1 + 2^2 + 2^2}\]
\[ = \sqrt{9}\]
\[ = 3\]
\[ \text{ Unit vector perpendicular to } \vec{a} \text{ and } \vec{b} =\frac{\vec{a} \times \vec{b}}{\left| \vec{a} \times \vec{b} \right|} = \frac{- \hat{ i } + 2 \hat{ j } + 2 \hat{ k } }{3}\]
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