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Question
Find the area of the triangle formed by O, A, B when \[\vec{OA} = \hat{ i } + 2 \hat{ j } + 3 \hat{ k } , \vec{OB} = - 3 \hat{ i } - 2 \hat{ j }+ \hat{ k } .\]
Solution
\[\text{ Given } : \]
\[ \vec{OA} = \hat{ i } + 2 \hat{ j } + 3 \hat{ k } \]
\[ \vec{OB} = - 3 \hat{ i } - 2 \hat{ j } + \hat{ k } \]
\[ \vec{OA} \times \vec{OB} = \begin{vmatrix}\hat{ i }& \hat{ j } & \hat{ k } \\ 1 & 2 & 3 \\ - 3 & - 2 & 1\end{vmatrix}\]
\[ = 8 \hat{ i }- 10 \hat{ j } + 4 \hat{ k } \]
\[\left| \vec{OA} \times \vec{OB} \right| = \sqrt{64 + 100 + 16}\]
\[ = \sqrt{180}\]
\[ = 6\sqrt{5}\]
\[\text{ Area of the triangle } =\frac{1}{2}\left| \vec{OA} \times \vec{OB} \right|\]
\[ =\frac{1}{2}\left( 6\sqrt{5} \right)\]
\[ =3\sqrt{5}\text{ sq. units } \]
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