Square of Vector Cross Product

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Theorem

Let $\mathbf a$ and $\mathbf b$ be vectors in a vector space $\mathbf V$ of $3$ dimensions.

Let $\mathbf a \times \mathbf b$ denote the vector cross product of $\mathbf a$ with $\mathbf b$.


Then:

$\paren {\mathbf a \times \mathbf b}^2 = \mathbf a^2 \mathbf b^2 - \paren {\mathbf a \cdot \mathbf b}^2$

where:

$\paren {\mathbf a \times \mathbf b}^2$ denotes the square of $\mathbf a \times \mathbf b$
$\mathbf a \cdot \mathbf b$ denotes the dot product of $\mathbf a \times \mathbf b$.


Proof 1

\(\ds \paren {\mathbf a \times \mathbf b}^2\) \(=\) \(\ds \paren {\mathbf a \times \mathbf b} \cdot \paren {\mathbf a \times \mathbf b}\) Definition of Square of Vector Quantity
\(\ds \) \(=\) \(\ds \paren {\mathbf a \cdot \mathbf a} \paren {\mathbf b \cdot \mathbf b} - \paren {\mathbf a \cdot \mathbf b} \paren {\mathbf b \cdot \mathbf a}\) Dot Product of Vector Cross Products
\(\ds \) \(=\) \(\ds \paren {\mathbf a \cdot \mathbf a} \paren {\mathbf b \cdot \mathbf b} - \paren {\mathbf a \cdot \mathbf b} \paren {\mathbf a \cdot \mathbf b}\) Dot Product Operator is Commutative
\(\ds \) \(=\) \(\ds \mathbf a^2 \mathbf b^2 - \paren {\mathbf a \cdot \mathbf b} \paren {\mathbf a \cdot \mathbf b}\) Definition of Square of Vector Quantity
\(\ds \) \(=\) \(\ds \mathbf a^2 \mathbf b^2 - \paren {\mathbf a \cdot \mathbf b}^2\) Definition of Square Function


Proof 2

The following proof is valid in the context of a Cartesian $3$-space:


Let $\theta$ denote the angle between $\mathbf a$ and $\mathbf b$

\(\ds \paren {\mathbf a \times \mathbf b}^2\) \(=\) \(\ds \paren {\mathbf a \times \mathbf b} \cdot \paren {\mathbf a \times \mathbf b}\) Definition of Square of Vector Quantity
\(\ds \) \(=\) \(\ds \paren {\norm {\mathbf a} \norm {\mathbf b} \sin \theta \, \hat {\mathbf n} } \cdot \paren {\norm {\mathbf a} \norm {\mathbf b} \sin \theta \, \hat {\mathbf n} }\) Definition 2 of Vector Cross Product: $\hat {\mathbf n}$ is a unit vector
\(\ds \) \(=\) \(\ds \paren {\norm {\mathbf a} \norm {\mathbf b} \sin \theta}^2\) Dot Product of Vector with Itself
\(\ds \) \(=\) \(\ds \paren {\norm {\mathbf a} \norm {\mathbf b} }^2 \paren {1 - \cos^2 \theta}\) Sum of Squares of Sine and Cosine
\(\ds \) \(=\) \(\ds \norm {\mathbf a}^2 \norm {\mathbf b}^2 - \paren {\norm {\mathbf a} \norm {\mathbf b} \cos \theta}^2\) rearranging
\(\ds \) \(=\) \(\ds \mathbf a^2 \mathbf b^2 - \paren {\mathbf a \cdot \mathbf b}^2\) Definition of Square of Vector Quantity, Definition of Dot Product
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