Question
Determine $\vec{a} \times \vec{b}$, given $\vec{a}=2 \hat{ i }+3 \hat{ j }$ and $\vec{b}=3 \hat{ i }+5 \hat{ j }$

Answer

Using determinant for vectors in two dimensions,
$\overrightarrow{ a } \times \overrightarrow{ b }=\left|\begin{array}{cc}\hat{ i } & \hat{ j } \\ a _x & a _y \\ b _x & b _y\end{array}\right|=\left|\begin{array}{cc}\hat{ i } & \hat{ j } \\ 2 & 3 \\ 3 & 5\end{array}\right|$
$
=[(2 \times 5)-(3 \times 3)] \hat{ k }=(10-9) \hat{ k }=\hat{ k }
$$
\vec{a} \times \vec{b} \text { gives } \hat{ k }
$

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