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Calculator Answers

Cross Product Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A cross product calculator computes the vector product of two three-dimensional vectors, yielding a third vector that is perpendicular to both inputs. The calculator evaluates the product using the determinant method of a three-by-three matrix, where the first row contains unit vectors, and subsequent rows contain the components of the input vectors. It determines the magnitude and direction of the resulting vector and can show the angle between the two input vectors. This tool is widely used in physics, engineering, and linear algebra to compute torque, angular momentum, and normal vectors.

Calculate the cross product A × B of two 3D vectors, with step-by-step determinant method.

Quick Answer

Compute the cross product of two 3D vectors. Enter the components of both vectors to find the resulting orthogonal vector, its magnitude, and the angle between them.

Vector A

Vector B

A × B Result

Cross Product Vector

(-3, 6, -3)

Orthogonal to both A and B

Magnitude |A × B|7.3485
Angle between A and B12.93°

Step-by-step Determinant

| i      j      k |

| 1 2 3 |

| 4 5 6 |

= i(2×6 - 3×5)

- j(1×6 - 3×4)

+ k(1×5 - 2×4)

= i(12 - 15) - j(6 - 12) + k(5 - 8)

= -3i + 6j - 3k

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How it works

What is the cross product?

The cross product is a mathematical operation on two vectors in three-dimensional space. It takes two vectors (A and B) and produces a third vector (C) that is completely perpendicular to both A and B.

The Determinant Method

The easiest way to remember how to calculate the cross product is to set it up as the determinant of a 3x3 matrix. The first row contains the unit vectors i, j, and k. The second row contains the components of vector A, and the third row contains the components of vector B.

| i j k |

| Ax Ay Az |

| Bx By Bz |

Expanding this determinant gives the components of the cross product:

  • x-component (i): Ay×Bz − Az×By
  • y-component (j): Az×Bx − Ax×Bz
  • z-component (k): Ax×By − Ay×Bx

Geometric Meaning

The cross product has two important geometric properties:

  • Direction: It is perpendicular to the plane containing the two original vectors. The exact direction is given by the right-hand rule.
  • Magnitude: Its length, |A × B|, is exactly equal to the area of the parallelogram formed by the vectors A and B.

Disclaimer

For educational purposes. Verify results for critical applications.

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Frequently asked questions

The cross product of two 3D vectors is a new vector that is perpendicular (orthogonal) to both of the original vectors. Its direction is determined by the right-hand rule, and its magnitude is equal to the area of the parallelogram that the two vectors span.

The cross product of A = (Ax, Ay, Az) and B = (Bx, By, Bz) is calculated as A × B = (Ay×Bz − Az×By, Az×Bx − Ax×Bz, Ax×By − Ay×Bx). It is often remembered by calculating the determinant of a 3x3 matrix containing the standard basis vectors i, j, k in the first row.

The right-hand rule helps you find the direction of the cross product A × B. If you point your right index finger in the direction of vector A, and your middle finger in the direction of vector B, your thumb will point in the direction of the cross product A × B.

No, the cross product is anticommutative. This means that A × B = −(B × A). Swapping the order of the vectors reverses the direction of the resulting vector.

If the cross product of two non-zero vectors is the zero vector, it means the two vectors are parallel or antiparallel (pointing in the same or exactly opposite directions). The angle between them is 0° or 180°.

Technically, the cross product is only defined in 3D (and 7D). However, for 2D vectors in the xy-plane, you can treat them as 3D vectors with a z-component of 0. Their cross product will be a vector pointing purely in the z-direction.

The cross product is used extensively in physics and engineering. Common applications include calculating torque, angular momentum, the magnetic force on a moving charge, and finding normal vectors to planes in computer graphics.

The magnitude of the cross product is related to the angle θ between the vectors by the formula |A × B| = |A| |B| sin(θ). You can use this to find the angle between two 3D vectors.