Cross Product Calculator

Free Online Cross Product Calculator

This cross product calculator instantly finds the vector cross product of two 3D vectors, A × B, along with the resultant vector's magnitude, the angle between the original vectors, and the area of the parallelogram and triangle they form. It works as a cross product calculator, vector product calculator, and 3D vector cross product calculator all in one tool, so you never need to work through the determinant expansion by hand.

Whether you're a student solving physics or linear algebra homework on torque, angular momentum, or surface normals, an engineer computing moments of force, or a graphics or game developer generating normal vectors for lighting and collision, this cross product calculator gives you an instant, accurate result along with a visual isometric diagram and complete step-by-step working, so the formula never feels like a black box.

How to Use This Cross Product Calculator

Using this online cross product calculator takes only a few seconds. Enter the x, y, and z components of vector A, then enter the x, y, and z components of vector B — the resultant vector, its magnitude, the angle between the vectors, and the parallelogram area update instantly as you type. The isometric diagram redraws automatically to show A, B, and A × B from the same origin, making the perpendicular relationship easy to visualize.

• Enter the x, y, and z components of vector A
• Enter the x, y, and z components of vector B
• Read the resultant vector A × B, its magnitude, and the angle between A and B
• Check the diagram and step-by-step solution to see exactly how each component was calculated

Cross Product Formula

The cross product (also called the vector product) of two 3D vectors produces a new vector that is perpendicular to both original vectors. For vectors A = (Ax, Ay, Az) and B = (Bx, By, Bz), each component of the result follows a cyclic pattern: skip the matching axis, then multiply and subtract the other two components in the reverse order. Enter any equation with numeric components, and the calculator solves every step automatically.

• x-component: Cx = Ay×Bz − Az×By
• y-component: Cy = Az×Bx − Ax×Bz
• z-component: Cz = Ax×By − Ay×Bx
• Magnitude formula: |A × B| = |A| × |B| × sin(θ)
• Area formula: Area of parallelogram = |A × B|, Area of triangle = |A × B| / 2

Worked Example: Finding the Cross Product Step by Step

Suppose A = (2, 3, 4) and B = (5, 6, 1). Start with the x-component: Cx = Ay×Bz − Az×By = (3)(1) − (4)(6) = 3 − 24 = −21. Next, the y-component: Cy = Az×Bx − Ax×Bz = (4)(5) − (2)(1) = 20 − 2 = 18. Finally, the z-component: Cz = Ax×By − Ay×Bx = (2)(6) − (3)(5) = 12 − 15 = −3.

So A × B = (−21, 18, −3), matching the default example above. Its magnitude is |A × B| = √((−21)² + 18² + (−3)²) = √(441 + 324 + 9) = √774 ≈ 27.82. This value also equals the area of the parallelogram formed by A and B, so the triangle formed by the same two vectors has an area of about 13.91.

The Right-Hand Rule and Direction of the Result

Unlike the dot product, which returns a single scalar number, the cross product always returns a vector — and that vector's direction matters just as much as its size. The direction of A × B is found using the right-hand rule: point the fingers of your right hand along vector A, curl them toward vector B through the smaller angle between them, and your thumb points in the direction of A × B.

This also explains why the cross product is anti-commutative: swapping the order reverses the direction, so B × A = −(A × B). If A and B are parallel (or one of them is the zero vector), the cross product is the zero vector, since there is no unique plane for a perpendicular direction to exist in.

Cross Product vs. Dot Product

The cross product and dot product both combine two vectors, but they measure completely different things. The dot product multiplies two vectors of any dimension and returns a single scalar related to how aligned the vectors are — largest when they point the same way, zero when perpendicular. The cross product only applies to 3D vectors and returns a brand-new vector perpendicular to both inputs, with a magnitude tied to the sine of the angle between them rather than the cosine.

A useful way to remember the difference: the dot product formula uses cos(θ) and tells you about similarity of direction, while the cross product formula uses sin(θ) and tells you about the perpendicular direction and the area swept between the two vectors. If you also need the scalar dot product, this calculator's sibling tool handles that separately.

Where the Cross Product Is Used

The cross product shows up anywhere a perpendicular direction, rotational effect, or swept area needs to be calculated from two vectors:

• Physics, calculating torque: Torque = Position vector × Force
• Physics, calculating angular momentum and magnetic force on a moving charge (F = qv × B)
• Computer graphics and game development, generating surface normal vectors for lighting and shading
• Engineering and robotics, finding the axis of rotation between two orientations
• Geometry, computing the area of a parallelogram or triangle directly from two edge vectors
• Aerospace and navigation, computing angular velocity and orbital mechanics quantities