Magnitude of Vector Cross Product equals Area of Parallelogram Contained by Vectors
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Theorem
Let $\mathbf a$ and $\mathbf b$ be vectors in a vector space of $3$ dimensions:
Let $\mathbf a \times \mathbf b$ denote the vector cross product of $\mathbf a$ with $\mathbf b$.
Then $\norm {\mathbf a \times \mathbf b}$ equals the area of the parallelogram two of whose sides are $\mathbf a$ and $\mathbf b$.
Proof
By definition of vector cross product:
- $\mathbf a \times \mathbf b = \norm {\mathbf a} \norm {\mathbf b} \sin \theta \, \mathbf {\hat n}$
where:
- $\norm {\mathbf a}$ denotes the length of $\mathbf a$
- $\theta$ denotes the angle from $\mathbf a$ to $\mathbf b$, measured in the positive direction
- $\mathbf {\hat n}$ is the unit vector perpendicular to both $\mathbf a$ and $\mathbf b$ in the direction according to the right-hand rule.
As $\mathbf {\hat n}$ is the unit vector:
- $\norm {\paren {\norm {\mathbf a} \norm {\mathbf b} \sin \theta \, \hat {\mathbf n} } } = \norm {\mathbf a} \norm {\mathbf b} \sin \theta$
By Area of Parallelogram, the area of the parallelogram equals the product of one of its bases and the associated altitude.
Let $\mathbf a$ denote the base of the parallelogram.
Then its altitude is $\norm {\mathbf b} \sin \theta$.
The result follows.
$\blacksquare$
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {II}$: The Products of Vectors: $5$. Vector Area
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 22$: Cross or Vector Product: $22.15$