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प्रश्न
if \[\vec{a} = \hat{ i }- 2\hat{ j } + 3 \hat{ k } , \text{ and } \vec{b} = 2 \hat{ i } + 3 \hat{ j } - 5 \hat{ k } ,\] then find \[\vec{a} \times \vec{b} .\] Verify th at \[\vec{a} \text{ and } \vec{a} \times \vec{b}\] are perpendicular to each other.
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उत्तर
\[\text{ Given } : \]
\[ \vec{a} = \stackrel\frown {i } - 2 \stackrel\frown {j } + 3 \stackrel\frown {k } \]
\[ \vec{b} = 2 \stackrel\frown{ i } + 3 \stackrel\frown{ j } - 5 \stackrel\frown {k } \]
\[\vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 1 & - 2 & 3 \\ 2 & 3 & - 5\end{vmatrix}\]
\[ = \hat{ i } + 11 \hat{ j } + 7 \hat{ k } \]
\[\text{ Now } ,\]
\[ \vec{a} . \left( \vec{a} \times \vec{b} \right) = 1 - 22 + 21\]
\[ = 0\]
\[\text{ Thus, } \vec{a} \text{ is perpendicular to } \vec{a} \times \vec{b} .\]
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