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Question
If \[\vec{r} = x \hat{ i } + y \hat{ j } + z \hat{ k } ,\] then write the value of \[\left| \vec{r} \times \hat{ i } \right|^2 .\]
Solution
\[\text{ Given } : \hspace{0.167em} \vec{r} = x \hat{ i } + y \hat{ j } + z \hat{ k } \]
\[\text{ Now } , \]
\[ \vec{i} = \hat { i } + 0 \hat{ j } + 0 \hat{ k } \]
\[ \vec{r} \times \vec{i} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ x & y & z \\ 1 & 0 & 0\end{vmatrix}\]
\[ = 0 \hat{ i } + z \hat{ j } - y \hat{ k } \]
\[ \Rightarrow \left| \vec{r} \times \vec{i} \right| = \sqrt{z^2 + y^2}\]
\[ \Rightarrow \left| \vec{r} \times \vec{i} \right|^2 = z^2 + y^2 \]
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