Skip to content

Maths 101: Back to Basics – Cross Product

May 20, 2013

Charles

So the cross product along with the dot product is the bread and butter of vector math – but I’d never known really whats happening internally. Essentially the cross product of two vectors (two directions), creates a new vector thats orthogonal to them. This means its a vector thats 90 degrees perpendicular to product of the other two.

So if we have two vectors [1,0,0] and [0,1,0], the cross product will be [0,0,1] – Likewise if we swap these vectors the cross product will be [0, 0, -1]. Even though we doing multiplication internally, we’re also doing subraction and vector order matters here. We can use sarrus’ rule, to get the cross product which is like finding the determinant – which i’ll discuss in matrix math:

If we have two vectors, [1, 0, 0] and [0, 1, 0] we can put them into a 3 by 3 matrix, with an imaginary row at the top:

I J K
1 0 0
0 1 0

We’ll get the determinant of each part of the imaginary row. Starting with the I, we’ll disregard any row and column it crosses and keep the rest. So for I it’ll become:

0 0
1 0

Next we’ll multiply the first value of the first row, by the last value of the last row – in this case 0, 0 and subtract it from first value of the last multiplied by the last value of the first row:

I = 1 * ((0 * 0) – (1 * 0)) = 0

This is the first part of a new vector – Lets see how this looks for the whole thing: cross ([1, 0, 0], [0,1,0]) =

I = 1 * ((0 * 0) – (1 * 0)) = 0
J = 1 * ((1 * 0) – (0 * 0)) = 0
K = 1 * ((1 * 1) – ( 0 * 0) = 1

We can see that the last part makes a difference, we’re doing (1 *1) – (0 * 0), so 1 – 0. If we’d have swapped the initial vectors around we’d have (0 * 0) – (1 * 1) = -1. Next up we’ll break into matrices..

Advertisements
No comments yet

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: