Vector Cross Product Calculator
Enter two 3-D vectors to compute the cross product a×b = (ay·bz−az·by, az·bx−ax·bz, ax·by−ay·bx), its magnitude |a×b|=|a||b|sinθ and the angle θ. a=(1,0,0), b=(0,1,0) → a×b=(0,0,1), |a×b|=1.
Input Data
Results
At a glance:The cross product (vector product) of two 3-D vectors a and b is a vector c=a×b that is perpendicular to both a and b, with magnitude equal to the area of the parallelogram they span: |c|=|a||b|sinθ, where θ is the angle between a and b. In components: c=(a_y·b_z−a_z·b_y, a_z·b_x−a_x·b_z, a_x·b_y−a_y·b_x). Direction follows the right-hand rule (curl fingers from a to b; thumb points along a×b). Key properties: (1) a×b=−(b×a) (anti-commutative); (2) a×a=0; (3) the result is perpendicular to both inputs (used for surface normals); (4) |a×b|=|a||b|sinθ is the parallelogram area — if a and b are parallel (sinθ=0), the area is zero. Example (default): a=(1,0,0), b=(0,1,0) → a×b=(0·0−0·1, 0·0−1·0, 1·1−0·0)=(0,0,1), magnitude 1, angle 90°. Applications: (1) torque τ=r×F; (2) angular momentum L=r×p; (3) magnetic force F=qv×B; (4) the normal vector of a plane (cross two edge vectors); (5) area of a triangle/parallelogram. Notes: (1) the cross product is defined only in 3-D (and 7-D); (2) order matters — a×b flips sign if swapped; (3) the magnitude is the parallelogram area, half of it for a triangle.
Formula
a×b = (a_y b_z − a_z b_y, a_z b_x − a_x b_z, a_x b_y − a_y b_x)
Magnitude: |a×b| = |a||b|sinθ
Right-hand rule sets the direction
Anti-commutative: a×b = −(b×a)
$$\mathbf{a}\times\mathbf{b} = (a_y b_z - a_z b_y,\ a_z b_x - a_x b_z,\ a_x b_y - a_y b_x), \quad |\mathbf{a}\times\mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta$$How to Use
- Enter the x, y, z components of vector a and vector b.
- The tool shows the three components of a×b, its magnitude and the angle θ.
- a×b is perpendicular to both a and b; a×a=0.
Case Studies
Magnetic force on a moving charge
Charge velocity v=(1,0,0) m/s, field B=(0,0,1) T, q=1 C.
F=qv×B = (0·1−0·0, 0·0−1·1, 1·0−0·0)=(0,−1,0) N.
Force points along −y, perpendicular to both v and B (right-hand rule).
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.