Using Vector Mathematics, Cross Products

Cross products are the most complicated use of vectors we've covered so far. One thing you'll need to know is if your game engine's coordinate system is left-handed or right-handed. If you're unfamiliar with coordinate system handedness, you might read over this article on Wikipedia.

The cross product method is very handy for finding the vector that is perpendicular/orthogonal to two other vectors. Any two non-parallel vectors will form a plane, and the line perpendicular to that plane that passes through the two vectors will always exist, that line can be found with a cross product. You might be asking how this is useful in 3D games? It's often used for finding an object's forward, up, or right vector. An object's forward vector is the vector that goes from the position of the object outward through the front of the object, an up vector is a vector that goes from the position of the object upward out of the top of the object, and the right vector goes from the position of the object out the right side of the object. Any vectors parallel to those vectors qualify as well. A picture might help clarify (or it might confuse you more):

Don't worry, this stuff confused me when I first started as well, a large part of being comfortable doing vector mathematics in games is just being able to picture multiple points and vectors in 3D, so you can picture exactly what it is you're trying to get. If you still want some info on Cross Product after this blog post, this video on YouTube might clarify.

Anyway, let's move on to explaining the math behind a cross product. Here's the exact formula:

If A and B are vectors, and x, y, z, denote the parameter within the vector

A (cross) B = ( Ay*Bz - Az*By, Az*Bx - Ax*Bz, Ax*By - Ay*Bx )

Don't feel bad if you don't memorize this, I haven't got it memorized either.

So let's take some vectors we know to be orthogonal, and see if the cross product gets us the same result:

Vector A = (1, 0, 0)

Vector B = (0, 1, 0)

Vector C = (0, 0, 1)

A (cross) B, should result in C given the three vectors we defined just above.

A (cross) B = ( 0*0 - 0*1, 0*0 - 1*0, 1*1 - 0*0 ) = (0, 0, 1)

As you can see A (cross) B, does indeed give us Vector C as we had defined it above.

And to give a real world scenario I often run into in actual game development. If we take an NPC that is always upright (by upright I mean that it's not laying on its side or something), if we want to get its right or left vectors, we just need the forward vector, and since its always upright we already have the up vector. Based on having the forward and up vector, forward (cross) up will give you the right vector.

Here's a guide of what vectors crossed (in a left-handed coordinate system) will give you:

Forward (cross) up --> Right

Up (cross) Forward --> Left

Forward (cross) right --> Down

Right (cross) Forward --> Up

Up (cross) right --> Backward

Right (cross) up --> Forward

If you read that first link I gave to Wikipedia about coordinate handedness, you'll find you can use your pointer finger, thumb, and middle finger to represent the three axes to determine the cross product order and what it will give you. 

I'll try and explain. Hold out your left hand in closed fist and give a thumbs up, now point your pointer finger straight ahead of you now point your middle finger out perpendicular to your thumb and pointer finger. In this position you can think of the cross product as:

Pointer (cross) Thumb --> Middle

Using this you can use whatever vectors you have to figure out which order to use the cross product. For example let's say you want an UP vector. Hold out your hand like I explained earlier, but turn your arm or wrist so that your middle finger is pointed upward now, and you will now see that your pointer finger is pointing Forward, and your thumb is pointed Left. Therefore: Forward (cross) Left --> Up, and you can always reverse the order to get Down, so Left (cross) Forward --> Down.

Hope this all makes sense. And here's the function for a cross product:

Vector3 Vector3::Cross(const Vector3& rhs) const
{
    return ( (y * rhs.z) - (z * rhs.y),
             (z * rhs.x) - (x * rhs.z),
             (x * rhs.y) - (y * rhs.x) );
}