Parallelogram area Triangle area A_ parallelogram=| a x b|=| a|| b|sintheta, A_ triangle= 12| a x b|. Choose two adjacent sides with the same starting point. For triangle ABC, use AB= b- a and AC= c- a, then halve the magnitude of their vector product. Position vectors b and c alone give the area of OBC, which may be a different triangle. Reversing the product changes its direction but not its magnitude or the area. Scaling the sides by k and m multiplies the area by |km|. A zero product means the sides are parallel and the triangle is degenerate.

\[A_{\rm parallelogram}=|\mathbf a\times\mathbf b|=|\mathbf a||\mathbf b|\sin\theta,\qquad A_{\rm triangle}=\frac12|\mathbf a\times\mathbf b|.\]

Areas

Choose two adjacent sides with the same starting point. For triangle \(ABC\), use \(\overrightarrow{AB}=\mathbf b-\mathbf a\) and \(\overrightarrow{AC}=\mathbf c-\mathbf a\), then halve the magnitude of their vector product. Position vectors \(\mathbf b\) and \(\mathbf c\) alone give the area of \(OBC\), which may be a different triangle.

Reversing the product changes its direction but not its magnitude or the area. Scaling the sides by \(k\) and \(m\) multiplies the area by \(|km|\). A zero product means the sides are parallel and the triangle is degenerate.

Areas from the Vector Product

Areas from the Vector Product

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