Cross Product Magnitude
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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Grade 12Grade 12 Math
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Cross Product Magnitude explained
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
- = Cross product magnitude
- = Magnitude of a
- = Magnitude of b
- = Angle between a and b (°)
Missing one of these? Work it out first, then come back
- Cross product magnitude — Cross Product z-Component of Two 2D Vectors
- Magnitude of a — Dot Product from Magnitudes and Included Angle, Resultant of Two Vectors at an Angle
- Magnitude of b — Dot Product from Magnitudes and Included Angle, Resultant of Two Vectors at an Angle
- Angle between a and b — Dot Product from Magnitudes and Included Angle, Resultant of Two Vectors at an Angle