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Question
If \[\vec{a} = \hat{ i } + \hat{ j } - \hat{ k } , \vec{b} = - \hat{ i } + 2\hat{ j } + 2 \hat{ k } \text{ and } \vec{c} = - \hat{ i } + 2 \hat{ j } - \hat{ k } ,\] then a unit vector normal to the vectors \[\vec{a} + \vec{b} \text{ and } \vec{b} - \vec{c}\] is
Options
\[\hat{ i } \]
\[\hat{ j } \]
\[\hat{ k } \]
none of these
Solution
\[\vec{a} + \vec{b} = 0 \hat{ i } + 3 \hat{ j } + \hat{ k } \]
\[ \vec{b} - \vec{c} = 0 \hat{ i }- 0 \hat { j } + 3 \hat{ k } \]
\[\left( \vec{a} + \vec{b} \right) \times \left( \vec{b} - \vec{c} \right) = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 0 & 3 & 1 \\ 0 & 0 & 3\end{vmatrix}\]
\[ = 9 \hat{ i } \]
\[\left| \left( \vec{a} + \vec{b} \right) \times \left( \vec{b} - \vec{c} \right) \right| = 9 \left| \hat{ i } \right|\]
\[ = 9\left( 1 \right)\]
\[ = 9\]
\[\text{ Unit vector perpendicular to both } \vec{a} + \vec{b} \text{ and } \vec{b} - \vec{c} = \frac{\left( \vec{a} + \vec{b} \right) \times \left( \vec{b} - \vec{c} \right)}{\left| \left( \vec{a} + \vec{b} \right) \times \left( \vec{b} - \vec{c} \right) \right|}\]
\[ = \frac{9 \hat{ i } }{9}\]
\[ = \hat{ i } \]
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