Calculate the vector product of two 3D vectors and find the perpendicular result
Cross Product (A × B)
Magnitude
📐 Formula: A × B = (a_y×b_z - a_z×b_y)i + (a_z×b_x - a_x×b_z)j + (a_x×b_y - a_y×b_x)k
The cross product (or vector product) of two 3D vectors produces a third vector perpendicular to both input vectors. Unlike the dot product (which gives a scalar), the cross product results in a vector.
Key Properties:
To determine the direction of A × B, use your right hand:
If you reverse the order (B × A), the result points in the opposite direction.
τ = r × F
Position vector crossed with force gives torque (rotational force). Direction indicates axis of rotation.
L = r × p
Position crossed with momentum. Conserved in closed systems, explaining why planets orbit in planes.
F = q(v × B)
Lorentz force on moving charge. Velocity crossed with magnetic field gives force perpendicular to both.
The magnitude of the cross product equals the area of the parallelogram formed by the two vectors:
|A × B| = |A| × |B| × sin(θ)
Where θ is the angle between vectors. When parallel (θ = 0°), sin(0) = 0, so cross product is zero. When perpendicular (θ = 90°), sin(90°) = 1, giving maximum magnitude.
The cross product can be calculated using a 3×3 determinant:
i-component: (2 × 6) - (3 × 5) = 12 - 15 = -3
j-component: (3 × 4) - (1 × 6) = 12 - 6 = 6
k-component: (1 × 5) - (2 × 4) = 5 - 8 = -3
✓ Result: A × B = (-3, 6, -3)
Magnitude: √(9 + 36 + 9) = √54 ≈ 7.35 (area of parallelogram formed by vectors)
No. The cross product is anti-commutative: A × B = -(B × A). Swapping the order reverses the direction of the result. The magnitude stays the same, but the direction flips 180°. This is why order matters in physics calculations like torque.
If vectors are parallel (or anti-parallel), their cross product is the zero vector [0, 0, 0]. This makes sense geometrically: parallel vectors form a degenerate parallelogram with zero area. Mathematically, sin(0°) = 0 or sin(180°) = 0, making |A × B| = 0.
Strictly speaking, no—cross product is defined only in 3D (and 7D, but that's exotic). However, you can treat 2D vectors as 3D with z = 0. The result is always (0, 0, k), pointing entirely along the z-axis. The k-value equals the signed area of the parallelogram (positive for counterclockwise, negative for clockwise).
Torque τ = r × F shows that maximum torque occurs when force is perpendicular to the lever arm (sin(90°) = 1). Pushing radially (toward/away from pivot) produces zero torque (sin(0°) = 0). The direction indicates the axis around which rotation occurs—clockwise vs counterclockwise depends on the cross product direction.
When expanding the 3×3 determinant along the first row, cofactor signs alternate: +i, -j, +k. This comes from the checkerboard pattern of determinant expansion. It's not an error—it's built into the mathematical structure. Without the negative j-term, the cross product wouldn't satisfy the right-hand rule.
Yes! A × (B + C) = (A × B) + (A × C). This distributive property is crucial in physics—forces can be decomposed into components, and torques calculated separately then summed. However, it's NOT associative: (A × B) × C ≠ A × (B × C). Order of operations matters!
Related Tools
Calculate spiral length.
Calculate earth circumference.
Calculate tensor product.
Calculate 3D distance.
Calculate vector addition.
Calculate vector operations.