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:
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)
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.
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