All calculatorsCross Product Calculator
Compute the cross product a × b for 2D or 3D vectors with exact fractions, optional step-by-step, magnitude, area (parallelogram/triangle), and a simple mini visual that shows direction (right-hand rule).
Background
The cross product takes two vectors and returns a new vector that is perpendicular to both. Its magnitude equals the area of the parallelogram formed by the vectors: |a × b| = |a||b|sin(θ). In 2D, we treat vectors as 3D with z = 0, so the result points along ±k.
How to use this calculator
- Choose 3D or 2D mode.
- Enter components for vectors a and b.
- Click Calculate to get a × b, its magnitude, and area.
- Optional: keep Prefer exact fractions on to avoid rounding issues.
Tip: If a × b = 0, the vectors are parallel (or one is the zero vector).
How this calculator works
- Uses the determinant form: a × b = ⟨aybz − azby, azbx − axbz, axby − aybx⟩
- Magnitude: |a × b| = √(cx² + cy² + cz²)
- Area: parallelogram = |a × b|, triangle = |a × b|/2
Formula & Equation Used
Cross product: a × b = ⟨aybz − azby, azbx − axbz, axby − aybx⟩
Magnitude / area: |a × b| = |a||b|sin(θ)
Example Problem & Step-by-Step Solution
Example 1 — 3D cross product
Let a = ⟨2, 3, 4⟩ and b = ⟨1, −2, 2⟩.
- Compute cx = aybz − azby = 3·2 − 4·(−2) = 14
- Compute cy = azbx − axbz = 4·1 − 2·2 = 0
- Compute cz = axby − aybx = 2·(−2) − 3·1 = −7
- So a × b = ⟨14, 0, −7⟩.
Example 2 — 2D (signed k-component)
Let a = ⟨3, 1⟩ and b = ⟨2, 5⟩ (treat z=0).
- Compute cz = axby − aybx = 3·5 − 1·2 = 13
- So a × b = ⟨0, 0, 13⟩ (points “out of the screen”).
- Area of parallelogram is |13| = 13.
Example 3 — Parallel vectors (cross = 0)
Let a = ⟨1, 2, 3⟩ and b = ⟨2, 4, 6⟩.
- Notice b = 2a, so the vectors are parallel.
- Parallel vectors form zero area, so a × b = ⟨0,0,0⟩.
- The unit normal direction is undefined because there’s no unique perpendicular direction.
Frequently Asked Questions
Q: What does the direction of a × b mean?
It points perpendicular to the plane of a and b using the right-hand rule. Swapping the order flips the sign: b × a = −(a × b).
Q: Why is my cross product zero?
If the vectors are parallel (or one is the zero vector), the parallelogram area is 0, so a × b = 0.
Q: What’s the geometric meaning of |a × b|?
It’s the area of the parallelogram formed by a and b (triangle area is half).
Q: Can I use fractions?
Yes. Turn on Prefer exact fractions to keep clean results and avoid rounding errors.
