In OpenGL vertex shaders, what is w, and why do I divide by it?

Solution 1:

W is the fourth coordinate of a three dimensional vertex; This vertex is called homogeneous vertex coordinate.

In few words, the W component is a factor which divides the other vector components. When W is 1.0, the homogeneous vertex coordinates are "normalized". To compare two vertices, you should normalize the W value to 1.0.

Think of the vertex (1,1,1,1). Now increase the W value (w > 1.0). The normalized position is scaling! and it is going to the origin. Think of the vertex (1,1,1,1). Now decrease the W value (W < 1.0). The normalized position is going to an infinite point.

Apart from scaling vertex coordinates, the W coordinate is necessary since you have to multiply a 4x4 matrix (the model view and/or the projection matrices) with a 4x1 matrix (the vertex).

Of course, the Red Book is the definite guide:

Red Book Appendix