These are vectors:
They can be multiplied using the "Cross Product" (also see Dot Product).
The Cross Product 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 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