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Question
If \[\vec{a} = a_1 \hat{ i } + a_2 \hat{ j } + a_3 \hat{ k } , \vec{b} = b_1 \hat{ i } + b_2 \hat{ j } + b_3 \hat{ k } \text{ and } \vec{c} = c_1 \hat{ i } + c_2 \hat{ j } + c_3 \hat{ k } ,\]then verify that \[\vec{a} \times \left( \vec{b} + \vec{c} \right) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c} .\]
Solution
\[\text{ Given } : \]
\[ \vec{a} = a_1 \hat{ i } + a_2 \hat{ j } + a_3 \hat{ k } \]
\[ \vec{b} = b_1 \hat{ i } + b_2 \hat{ j } + b_3 \hat{ k } \]
\[ \vec{c} = c_1 \hat{ i } + c_2\hat{ j } + c_3 \hat{ k } \]
\[ \vec{b} + \vec{c} = \left( b_1 + c_1 \right) \hat{ i } + \left( b_2 + c_2 \right) \hat{ j } + \left( b_3 + c_3 \right) \hat{ k } \]
\[ \therefore \vec{a} \times \left( \vec{b} + c \right) = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ a_1 & a_2 & a_3 \\ b_1 + c_1 & b_2 + c_2 & b_3 + c_3\end{vmatrix}\]
\[ = \left( a_2 b_3 + a_2 c_3 - a_3 b_2 - a_3 c_2 \right) \hat{ i } - \left( a_1 b_3 + a_1 c_3 - a_3 b_1 - a_3 c_1 \right) \hat{ j } + \left( a_1 b_2 + a_1 c_2 - a_2 b_1 - a_2 c_1 \right) \hat{ k } . . . (1)\]
\[\text{ Now } , \]
\[ \vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3\end{vmatrix}\]
\[ = \left( a_2 b_3 - a_3 b_2 \right) \hat{ i } - \left( a_1 b_3 - a_3 b_1 \right) \hat{ j } + \left( a_1 b_2 - a_2 b_1 \right) \hat{ k } \]
\[ \vec{a} \times \vec{c} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ a_1 & a_2 & a_3 \\ c_1 & c_2 & c_3\end{vmatrix}\]
\[ = \left( a_2 c_3 - a_3 c_2 \right) \hat{ i } - \left( a_1 c_3 - a_3 c_1 \right) \hat{ j } + \left( a_1 c_2 - a_2 c_1 \right) \hat{ k } \]
\[ \vec{a} \times \vec{b} + \vec{b} \times \vec{c} = \left( a_2 b_3 + a_2 c_3 - a_3 b_2 - a_3 c_2 \right) \hat{ i } - \left( a_1 b_3 + a_1 c_3 - a_3 b_1 - a_3 c_1 \right) \hat{ j } + \left( a_1 b_2 + a_1 c_2 - a_2 b_1 - a_2 c_1 \right) \hat{ k } . . . (2)\]
\[ \text{ From (1) and (2), we get } \]
\[ \vec{a} \times \left( \vec{b} + c \right) = \vec{a} \times \vec{b} + \vec{b} \times \vec{c}\]
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