Enter the components of two 3D vectors — Vector A (a₁, a₂, a₃) and Vector B (b₁, b₂, b₃) — and the Cross Product Calculator returns the resulting cross product vector (i, j, k components) along with the magnitude of the result. Perfect for physics, engineering, and linear algebra problems involving perpendicular vectors.
Magnitude |A × B|
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i component
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j component
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k component
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Magnitude |A|
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Magnitude |B|
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The cross product of two 3D vectors A and B is a third vector that is perpendicular to both A and B. It is defined as A × B = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). Unlike the dot product, the result is a vector, not a scalar.
Set up a 3×3 determinant with unit vectors i, j, k in the first row, Vector A components in the second row, and Vector B components in the third row. Expand along the first row: i(a₂b₃ − a₃b₂) − j(a₁b₃ − a₃b₁) + k(a₁b₂ − a₂b₁). Each cofactor gives you the i, j, and k components of the result.
The right-hand rule determines the direction of the cross product vector. Point your fingers in the direction of Vector A, curl them toward Vector B, and your thumb points in the direction of A × B. This is why the cross product is anti-commutative — A × B = −(B × A), because swapping the vectors reverses the direction.
The dot product of two vectors returns a scalar (a single number) representing how much the vectors align in the same direction. The cross product returns a new vector perpendicular to both input vectors and is related to the sine of the angle between them. The dot product is zero when vectors are perpendicular; the cross product is zero when they are parallel.
A zero cross product means the two vectors are parallel (or one of them is the zero vector). Since the magnitude of A × B equals |A| × |B| × sin(θ), when θ = 0° or 180° the sine is zero, producing a zero cross product vector.
The cross product is widely used in physics and engineering. Common applications include calculating torque (τ = r × F), finding the magnetic force on a moving charge (F = qv × B), computing the normal vector to a plane in 3D graphics, and determining angular momentum in rotational mechanics.
No, the cross product is anti-commutative, meaning A × B = −(B × A). Swapping the order of the vectors reverses the direction of the resulting vector. This is a key difference from scalar multiplication, where order does not matter.
The magnitude |A × B| equals the area of the parallelogram formed by vectors A and B. It is computed as |A| × |B| × sin(θ), where θ is the angle between the two vectors. This makes the cross product useful for computing areas in 3D space.