Cross Product

A vector has magnitude (how long it is) and direction:

vector magnitude and direction

Two vectors can be multiplied using the "Cross Product" (also see Dot Product)

vectors a and b

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

cross product

And it all happens in 3 dimensions!


We can calculate the Cross Product this way:

cross product with angle and unit vector

a × b = |a| |b| sin(θ) n

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:

cross product components

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

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

Answer: a × b = (−3,6,−3)


right hand rule

Which Direction?

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.



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

Answer: |elephant| |banana| sin(θ) n