Cross Product Magnitude
Magnitude of the vector perpendicular to two vectors.
Formula
Variables
Example
For perpendicular unit vectors, |a×b| = 1·1·sin90° = 1.
How it works
The cross product of two vectors produces a vector perpendicular to both, and this formula gives its length. The magnitude is the product of the side lengths times the sine of the included angle, so it measures the area of the parallelogram formed by the two vectors. It comes from the geometric fact that only the component of one vector perpendicular to the other contributes to this area; if the vectors are parallel, the sine is 0 and the cross product has zero magnitude.
When to use it
Use this when you need the area spanned by two vectors, or when you want to quantify how strongly two directions are non-parallel. It appears often in geometry, physics, and 3D vector calculations, especially for torques, normals, and surface area computations.
Worked example, step by step
Let |a| = 5, |b| = 8, and the angle between them be 30°. Use |a × b| = |a||b|sinθ. Substitute the values: |a × b| = 5 × 8 × sin(30°). Since sin(30°) = 1/2, we get |a × b| = 40 × 1/2. So, |a × b| = 20.
Common mistakes to avoid
- Using cosine instead of sine
- Forgetting that the angle must be the included angle between 0° and 180°
- Confusing the magnitude |a × b| with the vector a × b itself
Did You Know?
The cross product exists cleanly only in 3-D — it gives a vector perpendicular to both inputs.
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