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Question
If \[\vec{a}\] is any vector, then \[\left( \vec{a} \times \hat{ i } \right)^2 + \left( \vec{a} \times \hat{ j } \right)^2 + \left( \vec{a} \times \hat{ k } \right)^2 =\]
Options
- \[\vec{a}^2\]
\[2 \vec{a}^2\]
- \[3 \vec{a}^2\]
\[4 \vec{a}^2\]
Solution
\[2 \vec{a}^2\]
\[\text{ Let } \vec{a} = a_1 \hat{ i } + a_2 \hat{ j } + a_3 \hat{ k } \]
\[ \vec{a} \times \hat{ i } = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ a_1 & a_2 & a_3 \\ 1 & 0 & 0\end{vmatrix}\]
\[ = a_3 \hat{ j } - a_2 \hat{ k } \]
\[ \Rightarrow \left( \vec{a} \times \hat{ i } \right)^2 = \left( a_3 \hat{ j } - a_2 \hat{ k } \right)^2 \]
\[ = {a_3}^2 \left| \hat{ j } \right|^2 + {a_2}^2 \left| \hat{ k } \right|^2 - 2 a_3 a_2 \left( \hat{ j } . \hat{ k } \right)\]
\[ = {a_3}^2 + {a_2}^2 (\because \hat{ j } . \hat{ k } =0) . . . (1)\]
\[ \therefore \vec{a} \times \hat{ j } = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ a_1 & a_2 & a_3 \\ 0 & 1 & 0\end{vmatrix}\]
\[ = - a_3 \hat{ i } + a_1 \hat{ k } \]
\[ \Rightarrow \left( \vec{a} \times \hat{ j } \right)^2 = \left( - a_3 \hat{ i } + a_1 \hat{ k } \right)^2 \]
\[ = {a_3}^2 \left| \hat{ i } \right|^2 + {a_1}^2 \left| \hat{ k } \right|^2 - 2 a_3 a_2 \left( \hat{ i} . \hat{ k } \right)\]
\[ = {a_3}^2 + {a_1}^2 (\because \hat{ i } .\hat{ k } =0) . . . (2)\]
\[ \therefore \vec{a} \times \hat{ k } = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ a_1 & a_2 & a_3 \\ 0 & 0 & 1\end{vmatrix}\]
\[ = a_2 \hat{ i } - a_1 \hat{ j } \]
\[ \Rightarrow \left( \vec{a} \times k \right)^2 = \left( a_2 \hat{ i} - a_1 \hat{ j} \right)^2 \]
\[ = {a_2}^2 \left| \hat{ i } \right|^2 + {a_1}^2 \left| j \right|^2 + 2 a_1 a_2 \left( \hat{ i } . \hat{ j },\right)\]
\[ = {a_2}^2 + {a_1}^2 (\because \hat{ i } . \hat{ j } =0) . . . (3)\]
\[\text{ Adding (1), (2) and (3), we get } \]
\[ \left( \vec{a} \times \hat{ i } \right)^2 + \left( \vec{a} \times \hat{ j } \right)^2 + \left( \vec{a} \times k \right)^2 = {a_3}^2 + {a_2}^2 + {a_3}^2 + {a_1}^2 + {a_2}^2 + {a_1}^2 \]
\[ = 2 \left( {a_1}^2 + {a_2}^2 + {a_3}^2 \right)\]
\[ = 2 \vec{a}^2 (\because\left| \vec{a} \right|=\sqrt{{a_1}^2 + {a_2}^2 + {a_3}^2})\]
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