Cross Product

These are two vectors:

They can be multiplied using the "Cross Product"
(also see Dot Product).

The Cross Product a × b of two vectors is another vector that is at right angles to both:

And it all happens in 3 dimensions!


You can calculate the Cross Product this way:


a × b = |a| |b| sin(θ) n

|a| is the magnitude (length) of vector a

|b| is the magnitude (length) of vector b

θ is the angle between a and b

n is the unit vector at right angles to both a and b

So the length is: the length of a times the length of b times the sine of the angle between a and b,

Then you multiply by the vector n to make sure it heads in the right direction.

OR you can calculate it this way:


When a and b start at the origin point (0,0,0), the Cross Product will end at:

  • cx = aybz - azby
  • cy = azbx - axbz
  • cz = axby - aybx

They both work!


Example: What is the cross product of a = (2,3,4) and b = (5,6,7)


  • cx = aybz - azby = 3×7 - 4×6 = -3
  • cy = azbx - axbz = 4×5 - 2×7 = 6
  • cz = axby - aybx = 2×6 - 3×5 = -3


Answer: a × b = (-3,6,-3)


Which Way?

The cross product could point in the completely opposite direction and still be at right angles to the two other vectors, so we have the:

"Right Hand Rule"

With your right-hand, point your index finger along vector a, and point your middle finger along vector b: the cross product goes in the direction of your thumb.


Dot Product

The Cross Product gives a vector answer, and is sometimes called the "vector product"

But there is also the Dot Product which gives a scalar (ordinary number) as an answer.



Question: What do you get when you cross an elephant with a banana?

Answer: |elephant| |banana| sin(θ) n