Cross Product Calculator
Calculate the cross product of two three-dimensional vectors. Enter the components of a and b to get the vector a × b, its magnitude, the dot product and the angle between the two vectors.
- Magnitude |a × b|
- 7.348469
- Dot product a · b
- 56
- Angle between a and b
- 7.4758 °
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Formula
How to use it
- Type the three components of vector a, separated by commas.
- Type the three components of vector b.
- Read the cross product. Enter only two components for a vector in the x–y plane.
Worked examples
(2, 3, 4) × (5, 6, 7)
- a × b
- (-3, 6, -3)
- Magnitude |a × b|
- 7.348469
- Dot product a · b
- 56
- Angle between a and b
- 7.4758 °
The unit vectors along x and y
- a × b
- (0, 0, 1)
- Magnitude |a × b|
- 1
- Dot product a · b
- 0
- Angle between a and b
- 90 °
(3, −3, 1) × (4, 9, 2)
- a × b
- (-15, -2, 39)
- Dot product a · b
- -13
Properties of the cross product
The result is a vector perpendicular to both a and b, pointing the way given by the right-hand rule. Swapping the order flips its direction: b × a = −(a × b). If a and b are parallel, the cross product is the zero vector.
Its length equals the area of the parallelogram with sides a and b; half of that is the area of the triangle they form.
Where it is used
Torque is the cross product of the lever arm and the force. In 3-D graphics, the cross product of two edges of a triangle gives the surface normal used for lighting.
For two vectors in the x–y plane only the z component is non-zero, and it equals a₁b₂ − a₂b₁.
Questions people ask
What is the cross product of (2, 3, 4) and (5, 6, 7)?
(−3, 6, −3). Its magnitude is about 7.348.
What is i × j?
k. The unit vector along x crossed with the unit vector along y gives the unit vector along z: (1, 0, 0) × (0, 1, 0) = (0, 0, 1).
What is the difference between the cross product and the dot product?
The dot product is a single number that is largest when the vectors point the same way; the cross product is a vector that is largest when they are perpendicular.