Cross Product Calculator

The cross product takes two vectors in three-dimensional space and returns a third vector that is perpendicular to both. Its length equals the area of the parallelogram the two vectors span, and its direction follows the right-hand rule. This calculator returns the vector, its length, the unit normal, the parallelogram and triangle areas, the angle between the inputs, b × a, and a check that the result really is perpendicular to a and b.

You need it whenever a problem asks for a direction at right angles to a plane: the normal of a triangle in a 3D model, the axis of a rotation, the torque τ = r × F produced by a force on a lever, or the magnetic force F = qv × B on a moving charge. If you only need how much two vectors line up, use the dot product in the vector calculator instead.

For a = ⟨a₁, a₂, a₃⟩ and b = ⟨b₁, b₂, b₃⟩, expand a symbolic 3×3 determinant whose first row holds the unit vectors i, j and k:

a × b = (a₂b₃ − a₃b₂) i − (a₁b₃ − a₃b₁) j + (a₁b₂ − a₂b₁) k

Each component is a 2×2 determinant built from the other two coordinates. The middle term carries a minus sign from the cofactor pattern, and forgetting it is the most common hand error; the working panel shows that sign explicitly. The length of the result obeys

|a × b| = |a| |b| sin θ

where θ is the angle between a and b, from 0° to 180°. The calculator finds θ with atan2(|a × b|, a · b), which works across the whole range. Using arcsin alone, as some tools do, can only return angles up to 90°, so it reports 45° for vectors that are really 135° apart.

⟨1, 3, 4⟩ × ⟨2, 7, −5⟩

i: 3×(−5) − 4×7 = −15 − 28 = −43. j: −(1×(−5) − 4×2) = −(−13) = 13. k: 1×7 − 3×2 = 1. So a × b = ⟨−43, 13, 1⟩, with length 44.9333. Dotting the result with a gives 0, and with b gives 0, confirming it is perpendicular to both.

Area of a triangle in space

The triangle with vertices P(1, 4, 6), Q(−2, 5, −1) and R(1, −1, 1) has edge vectors PQ = ⟨−3, 1, −7⟩ and PR = ⟨0, −5, −5⟩. Their cross product is ⟨−40, −15, 15⟩, whose length is 45.2769 (exactly 5√82). The triangle is half the parallelogram, so its area is 22.6385 square units.

Torque on a wrench

A force of 20 N in the y direction applied 0.5 m along the x axis gives τ = r × F = ⟨1/2, 0, 0⟩ × ⟨0, 20, 0⟩ = ⟨0, 0, 10⟩ N·m. The torque has magnitude 10 N·m and points along +z, the axis the wrench turns about.

An obtuse angle

For ⟨1, 0, 0⟩ and ⟨−1, 1, 0⟩, a × b = ⟨0, 0, 1⟩ and |a × b| = 1, while |a||b| = √2, so sin θ = 0.7071. That sine fits both 45° and 135°. The dot product is −1, which is negative, so the angle is obtuse: θ = 135°.

Every cross product reduces to these nine results. Reading the table along the cycle i → j → k → i gives a positive result; going against it gives the negative.

• Anticommutative: b × a = −(a × b). Swapping the inputs swaps two rows of the determinant, which flips the sign.
• Zero for parallel vectors: if b is a multiple of a, or either vector is zero, the parallelogram has no area and a × b = 0. The calculator flags this case.
• Not associative: (a × b) × c is generally different from a × (b × c). For a volume, use the scalar triple product a · (b × c).
• Three dimensions only: the formula needs exactly three components. For 2D vectors, set the third component to 0; the z component of the result is then the signed area.
• Precision: results use double-precision arithmetic. Integer and fractional inputs give exact components; decimals can leave tiny rounding residues in the perpendicular check.