Cross Product of Perpendicular Vectors
Jump to navigation
Jump to search
Theorem
Let $\mathbf a$ and $\mathbf b$ be vector quantities which are perpendicular.
Let $\mathbf a \times \mathbf b$ denote the cross product of $\mathbf a$ with $\mathbf b$.
Then:
- $\mathbf a \times \mathbf b = \norm {\mathbf a} \norm {\mathbf b} \mathbf {\hat n}$
where:
- $\norm {\mathbf a}$ denotes the length of $\mathbf a$
- $\hat {\mathbf n}$ is the unit vector perpendicular to both $\mathbf a$ and $\mathbf b$ in the direction according to the right-hand rule.
Proof
By definition of cross product:
- $\norm {\mathbf a} \norm {\mathbf b} \sin \theta \, \mathbf {\hat n}$
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
- $\hat {\mathbf n}$ is the unit vector perpendicular to both $\mathbf a$ and $\mathbf b$ in the direction according to the right-hand rule.
When $\mathbf a$ and $\mathbf b$ are perpendicular, $\theta = 90 \degrees$ by definition.
Hence from Sine of Right Angle, $\sin \theta = 1$.
Hence the result.
$\blacksquare$
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {II}$: The Products of Vectors: $4$. The Vector Product: $(2.15)$