Monday, November 7, 2011

Using Vector Mathematics, Dot Products

Dot product is one of the properties of vectors that you can get when multiplying them. The dot product has many useful properties with vectors, in fact, the Dot Product and Cross Product are probably some of the most important properties of vectors in 3D math used in video games. The Dot Product is most often used to tell us information about the angle between two vectors, this can be used, for example, to tell if a point is in front of a player, or behind them.

To get a dot product, you multiply each component of two vectors, and then add each component together, like such:

Vector a = (3, -5, 7)
Vector b = (5, -2, -9)

a (dot) b = ((3 * 5) + (-5 * -2) + (7 * -9)) = 15 + 10 - 63 = -38

While the above method is using the dot product, most often you'll find you need the dot product of two unit-length vectors, which will always yield a result between -1 and +1, and as we'll discuss later on, that is extremely useful. So, we will unitize the two vectors from our above example, which will basically give us two direction vectors.

Length of a = √(3*3 + -5*-5 + 7*7) = √(9 + 25 + 49) = √83 = 9.110433
Inverse length of a = 1 / 9.110433  = 0.1097642
Unitized vector a = (3 * 0.1097642, -5 * 0.1097642, 7 * 0.1097642) = (0.3292926, -0.548821, 0.7683494)

Length of b = √(5*5 + -2*-2 + -9*-9) = √(25 + 4 + 81) = √110 = 10.488088
Inverse length of b = 1 / 10.488088 = 0.0953463
Unitized vector b = (5*0.0953463, -2*0.0953463, -9*0.0953463) = (0.4767315, -0.1906926, -0.8581167)

Alright, now that we have unit vectors let's try the dot product again:
a (dot) b = (0.3292926 * 0.4767315) + (-0.548821 * -0.1906926) + (0.7683494 * -0.8581167)
              = 0.1569842 + 0.1046561 + -0.65933345
              = -0.39769315, or let's just round it to -0.4

So, what does -0.4 really mean to you in this case? Well here's the cool part. With a dot product, if we treat the two vectors as directions, then a value less than 0 means that vector 'b' is behind vector 'a'. A value greater and 0 means it would be in front, and a value of 0 means it's directly to the left or right of 'a'.

So in this particular example, if we think of 'a' as the direction an object is facing, and think of vector 'b' as a point in space than that means that the point 'b' is behind the object. This is extremely useful for things like artificial intelligence, if an enemy knows the player is behind them, then it can be told to start turning around to face the player, or if the enemy knew the player was in front of them then they might know they are able to attack the player. Knowing where one object is in relation to another, especially based on that object's rotation, is essential for all kinds of things that require steering, not just A.I. So if you plan on making a 3D game, then the Dot Product is your friend. Learn it, live it, love it.

Here's a rough sketch of what this looks like, in this example I've done a top-down perspective, so I left the Y component out of the sketch, but based on the Y-components if this were in-game the enemy would be slightly higher than the player. Anyway, in the picture the arrow is the direction the player is facing, the line perpendicular to the arrow would be the left and right of the player, and as you can see the position of the enemy (which is 5, -9) is behind the player (and a bit to the right as well).

And finally, here's the code for our above example:

The Dot function would be a member of a Vector class most likely:
inline float Vector3::Dot( const Vector3& rhs ) const
    return (x * rhs.x
          + y * rhs.y
          + z * rhs.z);

And now to put our new Dot function to use:
Vector3 first = Vector3(3, -5, 7);
Vector3 second = Vector3(5, -2, -9);


float dotProduct = first.Dot(second);

// dotProduct is approx. -0.4

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