A vector perpendicular to two vectors
The cross product u × v is defined for 3D vectors. Unlike the dot product, it gives a vector, perpendicular to both u and v.
The formula
For u = [u₁, u₂, u₃] and v = [v₁, v₂, v₃]:
u × v = [u₂v₃ − u₃v₂, u₃v₁ − u₁v₃, u₁v₂ − u₂v₁]
It can also be found with a determinant:
| i | j | k |
|---|---|---|
| u₁ | u₂ | u₃ |
| v₁ | v₂ | v₃ |
Expanding along the top row gives the same result.
Example
u = [1, 2, 3], v = [4, 5, 6]
- x: 2(6) − 3(5) = 12 − 15 = −3
- y: 3(4) − 1(6) = 12 − 6 = 6
- z: 1(5) − 2(4) = 5 − 8 = −3
u × v = [−3, 6, −3]
Check: [−3, 6, −3] · [1, 2, 3] = −3 + 12 − 9 = 0 ✓ (perpendicular to u)
Direction: the right-hand rule
Point your right hand’s fingers along u, curl them toward v. Your thumb points in the direction of u × v. This is the right-hand rule.
i × j = k, j × k = i, k × i = j.
Order matters
v × u = −(u × v)
The cross product is anticommutative: swapping the order reverses the direction.
Magnitude and area
|u × v| = |u||v| sin θ
This magnitude equals the area of the parallelogram formed by u and v. The triangle they form has half that area.
Example: Area of the parallelogram from u = [1, 2, 3], v = [4, 5, 6]:
Area = |[−3, 6, −3]| = √(9 + 36 + 9) = √54 ≈ 7.35 square units
Parallel vectors
If u and v are parallel, u × v = 0 (sin 0° = 0).
Normal vectors
The cross product of two vectors in a plane gives a normal vector to the plane. This is key to writing equations of planes.
Application: torque
Torque (turning effect) = r × F, where r is the lever arm. Its magnitude |r||F| sin θ is greatest when the force is perpendicular to the lever. That’s why you push a door at the handle, at right angles.
Worked examples
Example 1: [1, 0, 0] × [0, 1, 0] = [0, 0, 1].
Example 2: [2, 1, 0] × [0, 3, 1] = [1·1 − 0·3, 0·0 − 2·1, 2·3 − 1·0] = [1, −2, 6].
Example 3: Area of a triangle with sides u = [2, 0, 0], v = [0, 3, 0]? |u × v| = 6 → triangle area 3.
Common mistakes
- Using the cross product in 2D (it’s a 3D operation).
- Sign errors in the middle component.
- Reversing the order and getting the opposite vector.
Try it yourself
- [1, 1, 0] × [0, 1, 1]
- [3, 0, 2] × [1, 4, 0]
- Area of the parallelogram from [1, 0, 0] and [1, 1, 0].
- What is u × u?
Answers: 1) [1, −1, 1] 2) [−8, 2, 12] 3) 1 4) the zero vector
Quick recap
- The cross product u × v is a vector perpendicular to both (3D only).
- Direction by the right-hand rule; v × u = −u × v.
- Magnitude = parallelogram area; it gives normal vectors and torque.