Cross Product Calculator

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Cross product mode gives the perpendicular vector, angle, and areas. Triple product mode adds a third vector and gives the parallelepiped volume.
Cross product a x bNon-parallel vectors
(-3, 6, -3)

The vector perpendicular to both a and b.

x-component-3
y-component6
z-component-3
Magnitude |a x b|7.348469
|a|3.741657
|b|8.774964
Angle between a and b12.9332deg
Parallelogram area7.348469
Triangle area3.674235
Unit normal x-0.408248
Unit normal y0.816497
Unit normal z-0.408248
x-component-3
y-component6
z-component-3

a x b = (-3, 6, -3) with magnitude 7.3485.

  • The result (-3, 6, -3) is perpendicular to both input vectors. Its dot product with each of them is exactly zero.
  • The magnitude 7.3485 equals the area of the parallelogram spanned by a and b. The triangle they form has half that area: 3.6742.
  • The angle between a and b is 12.9332 degrees. The cross product magnitude relates to this as |a x b| = |a||b|sin(theta).
  • Swapping the order to b x a flips every sign, giving a vector that points the opposite direction.

Next stepDivide the result by its magnitude to get the unit normal vector for the plane of a and b.

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