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!

Calculating

We 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 we multiply by the vector n to make sure it heads in the right direction (at right angles to both a and b).

 

OR we 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

Example: 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) answer, and is sometimes called the scalar product.

 

joke

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

Answer: |elephant| |banana| sin(θ) n