3D Maths for CG

Vectors - Dot Product

The dot product, aka inner product, takes two vectors and returns a scalar value. It is an easy way to find the cosine between two vectors. The equations bellow show the necessary steps to compute the inner product v, of two vectors v1 and v2. The operation is commonly represented by ".".

    	v1 . v2 = v1x*v2x + v1y*v2y + v1z*v2z

The relation between the cosine of the angle between v1 and v2 is given by

    	v1.v2 = |v1| |v2| * cos(a)

From the above we also know that the inner product of two vectors is 0 (zero) either if any of the vectors is null, or if the vectors are orthogonal, i.e. the angle is 90 degrees (see also the cross product). Bellow is a list of some properties of the inner product:

    	v1 . v2 = v2 . v1	

	v1 . (v2 + v3) = (v1 . v2) + (v1 . v3)

The inner product can be defined as a macro in C:

    	#define innerProduct(v,q) \
		((v)[0] * (q)[0] + \
		(v)[1] * (q)[1] + \
		(v)[2] * (q)[2])

[Previous: Cross Product] [Next: Vector Projection]

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