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Cross Product Magnitude

∣a⃗×b⃗∣=∣a⃗∣ ∣b⃗∣sin⁡θ|\vec{a}\times\vec{b}| = |\vec{a}|\,|\vec{b}|\sin\theta
sig figs

Worked example: |a| = 4, |b| = 5, 30° apart → |a×b| = 10. Press Try an example to run it live, then adjust anything.

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See also

Dot Product from Magnitudes and Included Angle: Both use vector magnitude and angle between vectors.Resultant of Two Vectors at an Angle: Both use vector magnitude and angle between vectors.Angle Between Two 2D Vectors (Components): Both use angle between vectors.Magnitude of a 2D Vector: Both use vector magnitude.Magnitude of a 3D Vector: Both use vector magnitude.Magnitude of a Scaled Vector: Both use vector magnitude.
4 more relatedScalar Projection of One Vector onto Another: Both use vector magnitude.Unit Vector Component (Normalization): Both use vector magnitude.x-Component from Magnitude and Angle: Both use vector magnitude.y-Component from Magnitude and Angle: Both use vector magnitude.
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Cross Product Magnitude explained

θ|a||b||a×b|

Where the dot product keeps the part of two vectors that agrees, the cross product keeps the part that disagrees: |a||b| sin θ, zero for parallel vectors and largest at a right angle. Numerically it is the area of the parallelogram the two vectors span, which is why torque, angular momentum and magnetic force all wear a cross product. With |a| = 4, |b| = 5 and 30° between them, |a × b| = 4 × 5 × 0.5 = 10.

The operation was born on 16 October 1843, when William Rowan Hamilton, after fifteen years of failing to multiply triples, walked along the Royal Canal in Dublin with his wife, saw the quaternion rule in a flash, and carved i² = j² = k² = ijk = −1 into the stone of Broom Bridge with a penknife. The vector part of a quaternion product is precisely the modern cross product; Gibbs and Heaviside later extracted it as a standalone operation. The trap when solving backwards for the angle is that arcsin returns only 0° to 90°: 30° and 150° give identical cross-product magnitudes, and only the sign of the dot product can tell you which of the two you have.

Cross Product Magnitude formula

∣a⃗×b⃗∣=∣a⃗∣ ∣b⃗∣sin⁡θ|\vec{a}\times\vec{b}| = |\vec{a}|\,|\vec{b}|\sin\theta
Where
  • ∣a⃗×b⃗∣|\vec{a}\times\vec{b}|= Cross product magnitude
  • ∣a⃗∣|\vec{a}|= Magnitude of a
  • ∣b⃗∣|\vec{b}|= Magnitude of b
  • θ\theta= Angle between a and b (°)
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Vector operations

12 formulasMagnitude, direction, components and unit vectors, plus the dot and cross products and the projections and angles they give you.

Revised 31 October 2026
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formula.expert. (2026, October 31). Cross Product Magnitude Formula & Calculator. https://formula.expert/solvers/cross-product-magnitude

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