Cross Product Calculator
Find the cross product of two 3D vectors.
Results update as you type.
How it works
The cross product combines two three-dimensional vectors to produce a third vector that is perpendicular to both, and this calculator computes it component by component along with its magnitude. You enter the x, y, and z components of vectors A and B, and it returns the resulting vector's three components and its length. The cross product is fundamental in physics and engineering: it gives the direction of torque, the force on a moving charge in a magnetic field, angular momentum, and the normal vector to a surface in computer graphics. Unlike the dot product, which yields a single number, the cross product yields a vector, and its direction is given by the right-hand rule: point your right hand's fingers from A toward B and your thumb points along the result. The magnitude of the cross product equals the area of the parallelogram formed by the two vectors, so it is zero when the vectors are parallel and largest when they are perpendicular. The cross product is defined only in three dimensions. Enter the components of both vectors to compute the cross product and its magnitude.
A × B = (ay·bz − az·by, az·bx − ax·bz, ax·by − ay·bx)
The cross product of two three-dimensional vectors produces a new vector perpendicular to both. Each component is a difference of products of the other two coordinates: the x-component uses the y and z values, and so on cyclically. The magnitude of the result equals the area of the parallelogram spanned by the two input vectors, and its direction follows the right-hand rule.
Source:Vector algebra
We review all our tools from time to time to ensure accuracy.
Worked example
How to use this calculator
- 1
Enter your values
Enter a: x, a: y, a: z, b: x, b: y and b: z.
- 2
Adjust the options
Fine-tune any options to match your situation.
- 3
Read your result
Result x (i) updates instantly as you type — no button to press.
Frequently asked questions
What is the cross product of two vectors?
The cross product of two three-dimensional vectors is a new vector perpendicular to both of the originals. Each of its components is computed from the other two coordinates of the input vectors. Unlike the dot product, the cross product returns a vector rather than a single number, and it exists only in three dimensions.
What does the magnitude of the cross product mean?
The magnitude of the cross product equals the area of the parallelogram formed by the two input vectors. It is zero when the vectors point in the same or opposite direction, since they span no area, and it reaches its maximum when the vectors are perpendicular. This makes the cross product useful for computing areas and testing for parallelism.
What is the right-hand rule?
The right-hand rule gives the direction of the cross product. Point the fingers of your right hand along the first vector, curl them toward the second, and your extended thumb points in the direction of the result. This is why the order matters: A cross B points opposite to B cross A.
How is the cross product different from the dot product?
The dot product multiplies two vectors to give a single scalar number related to how much they point in the same direction. The cross product gives a vector perpendicular to both, related to the area they span. The dot product works in any dimension, while the cross product is defined only in three dimensions.
Where is the cross product used?
The cross product appears throughout physics and engineering: torque is the cross product of position and force, the magnetic force on a moving charge uses it, and angular momentum is defined with it. In computer graphics it finds surface normals for lighting, and in geometry it computes areas and tests whether vectors are parallel.