Normalizing

A vector with a magnitude of 1 is a normal vector, sometimes called a unit vector. Whenever a vector has a length of 1, we can say that it has unit length. A normal vector is written as the letter of the vector with a caret symbol on top instead of an arrow, . We can normalize any vector by dividing each of its components by the length of the vector:

We never implemented division operators for the vector class. We can rewrite the preceding equation as reciprocal multiplication. This means we can obtain the normal of a vector if we multiply that vector by the inverse of its length:

Getting ready

We are going to implement two functions, Normalize and Normalized. The first function will change the input vector to have a length of 1. The second function will not change the input vector; rather it will return a new vector with a length of 1.

How to do it…

Follow these steps to implement functions which will make a vector unit length or return a unit length vector. These steps utilize reciprocal multiplication.

1. Declare the Normalize and Normalized functions in vectors.h:

   void Normalize(vec2& v);
   void Normalize(vec3& v);
   
   vec2 Normalized(const vec2& v);
   vec3 Normalized(const vec3& v);

2. Add the implementation of these functions to vectors.cpp:

   void Normalize(vec2& v) {
      v = v * (1.0f / Magnitude(v));
   }
   
   void Normalize(vec3& v) {
      v = v * (1.0f / Magnitude(v));
   }
   
   vec2 Normalized(const vec2& v) {
      return v * (1.0f / Magnitude(v));
   }
   
   vec3 Normalized(const vec3& v) {
      return v * (1.0f / Magnitude(v));
   }

How it works…

Normalizing works by scaling the vector by the inverse of its length. This scale makes the vector have unit length, which is a length of 1. Unit vectors are special as any number multiplied by 1 stays the same number. This makes unit vectors ideal for representing a direction. If a direction has unit length, scaling it by some velocity becomes trivial.