The force that twists instead of pushes
Push on a wrench handle and the bolt turns: the twisting effect, called torque, points along the bolt, perpendicular to both your push and the handle. That perpendicular direction is exactly what the cross product computes. It takes two 3D vectors and returns a third vector at right angles to both.
The cross product formula
Formula: a x b = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)
Each component mixes two coordinates from a and two from b in a fixed pattern. The result is perpendicular to both inputs, and its length equals the area of the parallelogram they span. Reversing the order flips the sign: b x a = -(a x b).
Using the calculator
- Enter the three components of the first vector.
- Enter the three components of the second vector.
- Read the resulting perpendicular vector.
A worked example, step by step
Compute (1, 2, 3) x (4, 5, 6).
- Step 1: first component: 2x6 - 3x5 = 12 - 15 = -3.
- Step 2: second component: 3x4 - 1x6 = 12 - 6 = 6.
- Step 3: third component: 1x5 - 2x4 = 5 - 8 = -3.
- Step 4: verify it is perpendicular. Dot with (1, 2, 3): -3 + 12 - 9 = 0. Dot with (4, 5, 6): -12 + 30 - 18 = 0.
The cross product is (-3, 6, -3), and both dot products confirm it sits at right angles to the inputs.
Where cross products matter
Physics uses them for torque and magnetic force; 3D graphics uses them to find surface normals for lighting. For the steepness of a line instead, see the slope calculator, and for the distance between two points, the distance formula calculator.
Mistakes that give the wrong answer
- Swapping the order. The cross product is not commutative: reversing the inputs negates every component.
- Feeding it parallel vectors. Parallel inputs produce the zero vector, which carries no direction at all.
- Mixing up component positions. The pattern of subscripts is easy to scramble by hand; the calculator follows it exactly.
The length is an area
The cross product carries a bonus fact: its length equals the area of the parallelogram spanned by the two inputs. For (1, 2, 3) x (4, 5, 6) = (-3, 6, -3), the length is sqrt(9 + 36 + 9) = sqrt(54), about 7.35. Engineers use this to compute the area of parallelogram plates, and physicists read torque magnitude straight from it: the twisting strength of a force about a point is exactly this area formed by the position and force vectors.
What this assumes
The cross product is defined only for three-dimensional vectors. In two dimensions or four, the same idea needs different tools.