Mathematical Prerequisites - Introduction to 3D Game Programming with DirectX 9.0 - page 6

Similarly, we can test if two vectors are not equal using the overloaded
not equals operator:
if( u != v ) return true;

Note: When comparing floating-point numbers, care must be taken
because, due to floating-point imprecision, two floating-point numbers
that we expect to be equal may differ slightly; therefore, we test if they
are approximately equal. We do this by defining an EPSILON constant,
which is a very small value that we use as a “buffer.” We say two val-
ues are approximately equal if their distance is less than EPSILON .In
other words, EPSILON gives us some tolerance for floating-point
imprecision. The following function illustrates how EPSILON can be
used to test if two floating-point values are equal:
const float EPSILON = 0.001f;
bool Equals(float lhs, float rhs)
{

// if lhs == rhs their difference should be zero
return fabs(lhs - rhs) < EPSILON ? true : false;

}
We do not have to worry about doing this when using the D3DXVEC-
TOR3 class, as its overloaded comparison operations will do this for us,
but comparing floating-point numbers properly is important to know in
general.

Computing the Magnitude of a Vector
Geometrically, the magnitude of a vector is the length of the directed
line segment. Given the components of a vector, we can algebraically
compute its magnitude with the following formula:

(1)

2 2 2

u u u u

x y z

The double vertical bars in u denotes the magnitude of u .

Example: Find the magnitude of the vectors u = (1, 2, 3) and v = (1, 1).
Solution:For u we have:

(2)

2 2 2

u 1 2 3 1 4 9 14

Generalizing formula (1) to two dimensions, for v we have:

2 2

(3)

v 1 1 2

Using the D3DX library, we can compute the magnitude of a vector
using the following function: