Quick Answer

The cross product of (1, 2, 3) and (4, 5, 6) is (-3, 6, -3) with a magnitude of approximately 7.35.

Vector A

a_x

a_y

a_z

Vector B

b_x

b_y

b_z

Results

Common Examples

Input	Result
(1, 0, 0) x (0, 1, 0)	(0, 0, 1), magnitude 1
(1, 2, 3) x (4, 5, 6)	(-3, 6, -3), magnitude 7.35
(2, 4, 6) x (1, 2, 3)	(0, 0, 0), parallel vectors
(3, -1, 2) x (1, 4, -2)	(-6, 8, 13), magnitude 16.16

How It Works

The formula

The cross product of two 3D vectors produces a vector perpendicular to both inputs. For vectors \(\mathbf{A} = (a_1, a_2, a_3)\) and \(\mathbf{B} = (b_1, b_2, b_3)\):

\[\mathbf{A} \times \mathbf{B} = (a_2 b_3 - a_3 b_2,\ a_3 b_1 - a_1 b_3,\ a_1 b_2 - a_2 b_1)\]

This can also be written as the determinant of a 3x3 matrix with unit vectors i, j, k in the first row.

The magnitude of the cross product equals the area of the parallelogram formed by the two vectors:

\[|\mathbf{A} \times \mathbf{B}| = |\mathbf{A}||\mathbf{B}|\sin\theta\]

where \(\theta\) is the angle between the vectors.

Key properties

• The cross product is anticommutative: A x B = -(B x A)
• Parallel vectors have a cross product of zero (sin 0 = 0)
• Perpendicular unit vectors have a cross product magnitude of 1
• The result is always perpendicular to both input vectors (right-hand rule)

Worked example

For A = (1, 2, 3) and B = (4, 5, 6):

• x-component: (2)(6) - (3)(5) = 12 - 15 = -3
• y-component: (3)(4) - (1)(6) = 12 - 6 = 6
• z-component: (1)(5) - (2)(4) = 5 - 8 = -3

Result: (-3, 6, -3). Magnitude = sqrt(9 + 36 + 9) = sqrt(54) = 7.35.

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Frequently Asked Questions

What does the cross product represent geometrically?

The cross product of two vectors produces a third vector that is perpendicular to both input vectors. Its magnitude equals the area of the parallelogram formed by the two vectors. The direction follows the right-hand rule: point your fingers along vector A, curl them toward vector B, and your thumb points in the direction of A x B.

When is the cross product zero?

The cross product is zero when the two vectors are parallel (pointing in the same or opposite directions) or when either vector is the zero vector. Parallel vectors have an angle of 0 or 180 degrees between them, and sin(0) = sin(180) = 0.

Can you compute a cross product in 2D?

The cross product is defined for 3D vectors only. For 2D vectors, you can treat them as 3D vectors with a zero z-component. The result will be a vector pointing entirely in the z-direction, and its magnitude represents the signed area of the parallelogram formed by the two 2D vectors.

What is the difference between cross product and dot product?

The dot product of two vectors produces a scalar (a single number), while the cross product produces a new vector. The dot product measures how much two vectors point in the same direction. The cross product measures the perpendicular component and produces a vector orthogonal to both inputs.

Why is the cross product anticommutative?

Switching the order of the two vectors reverses the direction of the result: A x B = -(B x A). This is because the right-hand rule gives opposite directions when you switch the order of the inputs. The magnitude stays the same, but the resulting vector points in the opposite direction.