Square of Norm of Vector Cross Product

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Theorem

Let $\mathbf a$ and $\mathbf b$ be vectors in the real Euclidean space $\R^3$.

Let $\times$ denote the vector cross product.


Then:

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


Proof

Let $\mathbf a = \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix}$, and $\mathbf b = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}$.

Then:

\(\ds \norm {\mathbf a \times \mathbf b }^2\) \(=\) \(\ds \paren {\mathbf a \times \mathbf b } \cdot \paren {\mathbf a \times \mathbf b}\) Definition of Euclidean Norm
\(\ds \) \(=\) \(\ds \begin {bmatrix} a_2 b_3 - a_3 b_2 \\ a_3 b_1 - a_1 b_3 \\ a_1 b_2 - a_2 b_1 \end {bmatrix} \cdot \begin {bmatrix} a_2 b_3 - a_3 b_2 \\ a_3 b_1 - a_1 b_3 \\ a_1 b_2 - a_2 b_1 \end {bmatrix}\) Definition of Vector Cross Product
\(\ds \) \(=\) \(\ds {a_2}^2 {b_3}^2 + {a_3}^2 {b_2}^2 - 2 a_2 a_3 b_2 b_3 + {a_3}^2 {b_1}^2 + {a_1}^2 {b_3}^2 - 2 a_1 a_3 b_1 b_3 + {a_1}^2 {b_2}^2 + {a_2}^2 {b_1}^2 - 2 a_1 a_2 b_1 b_2\) Definition of Dot Product
\(\ds \) \(=\) \(\ds \paren { {a_1}^2 + {a_2}^2 + {a_3}^2} \paren { {b_1}^2 + {b_2}^2 + {b_3}^2} - \paren {a_1 b_1 + a_2 b_2 + a_3 b_3}^2\) by algebraic manipulations
\(\ds \) \(=\) \(\ds \paren {\mathbf a \cdot \mathbf a} \paren {\mathbf b \cdot \mathbf b} - \paren {\mathbf a \cdot \mathbf b}^2\) Definition of Dot Product
\(\ds \) \(=\) \(\ds \norm {\mathbf a}^2 \norm {\mathbf b}^2 - \paren {\mathbf a \cdot \mathbf b}^2\) Definition of Euclidean Norm

$\blacksquare$


Sources

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