Find Cross Products Calculator

Enter the components of two 3D vectors to calculate their cross product.

Vector A

Vector B

Chart comparing the magnitudes of Vector A, Vector B, and the Cross Product Vector C.

Vector	x-component	y-component	z-component	Magnitude
A	1	2	3	—
B	4	5	6	—
C (A x B)	—	—	—	—

Table summarizing the components and magnitudes of the input vectors and their cross product.

What is a Cross Product Calculator?

A cross product calculator is a tool used to determine the cross product (or vector product) of two vectors in three-dimensional space. The cross product of two vectors, say A and B, results in a third vector, C, which is perpendicular (orthogonal) to the plane containing A and B. The direction of C is given by the right-hand rule, and its magnitude is related to the area of the parallelogram formed by A and B.

This calculator is particularly useful for students, engineers, physicists, and computer graphics professionals who frequently work with vectors and need to find a vector normal to a plane, calculate torque, or understand the relationship between two vectors in 3D space. Our cross product calculator simplifies these calculations.

Common misconceptions include thinking the cross product is commutative (A x B = B x A, which is false; A x B = -B x A) or that it’s the same as the dot product (which results in a scalar, not a vector).

Cross Product Formula and Mathematical Explanation

The cross product of two vectors A = (A_x, A_y, A_z) and B = (B_x, B_y, B_z) is defined as:

A x B = (A_yB_z – A_zB_y)i + (A_zB_x – A_xB_z)j + (A_xB_y – A_yB_x)k

where i, j, and k are the standard unit vectors along the x, y, and z axes, respectively. The components of the resulting vector C = (C_x, C_y, C_z) are:

• C_x = A_yB_z – A_zB_y
• C_y = A_zB_x – A_xB_z
• C_z = A_xB_y – A_yB_x

This can also be expressed as the determinant of a matrix:

A x B = | i   j   k |
            | A_x A_y A_z |
            | B_x B_y B_z |

The magnitude of the cross product is given by |A x B| = |A| |B| sin(θ), where θ is the angle between A and B, and |A| and |B| are their magnitudes. The cross product calculator computes these components and the magnitude for you.

Variables Table

Variable	Meaning	Unit	Typical Range
Ax, Ay, Az	Components of vector A	Varies (length, velocity, force, etc.)	-∞ to +∞
Bx, By, Bz	Components of vector B	Varies (length, velocity, force, etc.)	-∞ to +∞
Cx, Cy, Cz	Components of the cross product vector C	Depends on units of A and B	-∞ to +∞
|C|	Magnitude of the cross product vector C	Depends on units of A and B	0 to +∞
i, j, k	Unit vectors along x, y, z axes	Dimensionless	Unit length

Practical Examples (Real-World Use Cases)

Example 1: Finding a Normal Vector

Suppose we have two vectors lying on a plane: A = (2, 1, -1) and B = (1, 3, 2). To find a vector perpendicular to this plane, we calculate their cross product using our cross product calculator or manually:

• C_x = (1)(2) – (-1)(3) = 2 + 3 = 5
• C_y = (-1)(1) – (2)(2) = -1 – 4 = -5
• C_z = (2)(3) – (1)(1) = 6 – 1 = 5

So, the cross product C = (5, -5, 5). This vector is orthogonal to both A and B.

Example 2: Calculating Torque

Torque (τ) is the rotational equivalent of linear force and is defined as the cross product of the position vector (r) from the axis of rotation to the point where the force is applied, and the force vector (F): τ = r x F. If r = (1, 1, 0) meters and F = (0, 10, 0) Newtons:

• τ_x = (1)(0) – (0)(10) = 0
• τ_y = (0)(0) – (1)(0) = 0
• τ_z = (1)(10) – (1)(0) = 10

The torque vector is τ = (0, 0, 10) Newton-meters, meaning a torque of 10 Nm around the z-axis. The cross product calculator can easily find this.

How to Use This Cross Product Calculator

1. Enter Vector A Components: Input the x, y, and z components (A_x, A_y, A_z) of the first vector into the respective fields under “Vector A”.
2. Enter Vector B Components: Input the x, y, and z components (B_x, B_y, B_z) of the second vector into the respective fields under “Vector B”.
3. View Real-Time Results: The calculator automatically computes the cross product vector C = (C_x, C_y, C_z), its magnitude |C|, and the magnitudes of A and B as you type. The primary result shows the vector C, and intermediate results display individual components and magnitudes.
4. Interpret the Output: The primary result is the vector C = A x B. The intermediate values give its components and the magnitudes of all three vectors. The table and chart also summarize this information.
5. Reset: Click the “Reset” button to clear all inputs and results and return to the default values.
6. Copy Results: Click “Copy Results” to copy the calculated vector C, its components, and magnitudes to your clipboard.

This cross product calculator is designed for ease of use, providing instant and accurate results.

Key Factors That Affect Cross Product Results

• Vector Components: The individual x, y, and z components of both input vectors directly determine the components and magnitude of the cross product. Changing any component will alter the result.
• Magnitude of Vectors: The magnitude of the cross product is proportional to the product of the magnitudes of the original vectors (|A x B| = |A| |B| sin(θ)). Larger vectors tend to result in a larger cross product magnitude if the angle is not zero.
• Angle Between Vectors (θ): The sine of the angle between the two vectors is a crucial factor. The cross product’s magnitude is maximum when the vectors are perpendicular (sin(90°) = 1) and zero when they are parallel or anti-parallel (sin(0°) = sin(180°) = 0).
• Order of Vectors: The cross product is anti-commutative (A x B = -B x A). Swapping the order of the vectors will negate the resulting vector (it will point in the opposite direction but have the same magnitude). Our cross product calculator respects this order.
• Coordinate System: The components and the resulting cross product depend on the chosen coordinate system (e.g., right-handed Cartesian).
• Right-Hand Rule: The direction of the cross product vector C is determined by the right-hand rule applied to vectors A and B.
• Units of Components: The units of the cross product’s components and magnitude depend on the units of the components of the input vectors. If A and B represent displacements (meters), the cross product’s magnitude relates to area (square meters). If A is position (m) and B is force (N), the result is torque (Nm).

Frequently Asked Questions (FAQ)

Is the cross product commutative?

No, A x B = – (B x A). The order matters, and reversing it changes the direction of the resulting vector.

What is the cross product of a vector with itself?

The cross product of any vector with itself is the zero vector (0, 0, 0), because the angle between them is 0, and sin(0) = 0.

What is the geometric meaning of the cross product’s magnitude?

The magnitude of A x B, |A x B|, is equal to the area of the parallelogram formed by vectors A and B when they are placed tail-to-tail.

When is the cross product zero?

The cross product is zero if one or both vectors are the zero vector, or if the vectors are parallel or anti-parallel (the angle between them is 0° or 180°).

What’s the difference between the cross product and the dot product?

The cross product (A x B) results in a vector perpendicular to both A and B, while the dot product (A · B) results in a scalar (a number) related to the projection of one vector onto another. Check our dot product calculator.

How is the cross product used in physics?

It’s used to calculate torque (τ = r x F), angular momentum (L = r x p), the magnetic force on a moving charge (F = q(v x B)), and to find normal vectors.

Can I use the cross product calculator for 2D vectors?

The cross product is inherently defined for 3D vectors. To find something analogous in 2D, you can represent 2D vectors as 3D vectors with z-components set to zero (e.g., (x, y) becomes (x, y, 0)). The cross product will then be a vector along the z-axis.

How do I find the angle between two vectors using the cross product?

You can use the formula |A x B| = |A| |B| sin(θ). So, sin(θ) = |A x B| / (|A| |B|). You’d also need the dot product to resolve the quadrant of the angle as sin(θ) = sin(180°-θ).