Computing the Cross ProductThe formula to compute the cross product is a complex one when vectors are in Cartesian form: if u = [a, b, c] and v = [d, e, f] then the cross product uxv = [bf - ce, cd - af, ae – bd] That’s a lot of letters to remember.To help this process, mathematicians have come up with different ways to simplify it. Here is one.

1. First, write out each vector in a column in the following way:

a d

b e

c f

a d

b e

The first column comes from the first vector in the cross product. Notice the last two values in the column are just the first two repeated again.

The second column comes from the second vector and similarly the last two values are the first two repeated.

Computing the Cross Product2. Now box out the first row. We won’t be needing these a d

b e

c f

a d

b e

3. Now draw arrows going down from the left and up to the right.

4. Each set of arrows define one of the variables (x, y, z). And each set of arrows represent a difference of products: down products minus up products

x = bf - ce

y = cd - af

z = ae - bd

Down - up

5. Now write out the answer: uxv = [bf - ce, cd - af, ae – bd]

Computing the Cross Product

1 -4

-2 53 61 -4-2 5

x = -2(6) – 3(5)

y = 3(-4) –1(6)

z=1(5)–(-2)(-4)

Down - up

Lets try one: [1, -2,3] x [-4,5,6] Thus [1, -2,3] x [-4,5,6] = [-27, -18, -3]

How can we check our answer?

Use the fact that the vector produced by the cross product is perpendicular to both of the original vectors.

What does this mean?

The dot product of the answer with either of the original vectors will be zero.

[1, -2,3]·[-27, -18, -3] = 1(-27) + (-2)(-18) + 3(-3)= -27 + 36 – 9 = 0

Thus it checks out.