MathGem:Vector Operations

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Contents

[edit] Symbols

A brief overview of mathematical symbols relevant to this page.


\vec u, \vec v, \vec w - common generic vectors

\begin{bmatrix} x \\ y \\ z \end{bmatrix} - visual representation of a vector

\hat u, \hat v, \hat w - generic unit vectors (length = 1)

\left | \vec v \right | - length of a vector

θ - angle between two vectors

\vec v \cdot \vec w - dot product (of \vec v and \vec w)

\vec v \times \vec w - cross product (of \vec v and \vec w)


[edit] Length Of A Vector

A simple yet powerful operation, lengths of a vector are used in a variety of the more complex vector matrix operations. Also, they can be used to determine the distance between two objects, by creating a vector from the subtraction of the two objects' coordinates and then taking the length of that vector:

\left | \vec v \right | = \sqrt{{\vec v}_x^{~2} + {\vec v}_y^{~2} + {\vec v}_z^{~2}}

[edit] C

#include <math.h>
 
typedef struct Vector3_
{
  double x;
  double y;
  double z;
} Vector3;
 
double length(Vector3 *v)
{
  return (sqrt(v->x*v->x + v->y*v->y + v->z*v->z));
}

[edit] C++

#include <cmath>
 
class Vector3
{
public:
	// Members
	double x, y, z;
 
	// Constructors
	Vector3() { } 
	Vector3( const Vector3 &r ) : x(r.x), y(r.y), z(r.z) { }
	Vector3( const double x, const double y, const double z ) : x(x), y(y),	z(z) { }
	~Vector3() { }
 
	// Methods
	double length() const
	{
		return std::sqrt( x * x + y * y + z * z );
	}
};

Note that there's a Complete math::vector Class elsewhere on the wiki that you can use instead of writing your own.

[edit] C#

public class Vector3
{
  public double X;
  public double Y;
  public double Z;
 
  public double Length
  {
    get{return System.Math.Sqrt(X*X + Y*Y + Z*Z);}
  }
}

[edit] Python

# v is a tuple representing a 3d vector
def length(v):
    return (v[0]*v[0] + v[1]*v[1] + v[2]*v[2]) ** 0.5

[edit] Visual Basic

' In Public Object Module:
Public Type Vector3
  X As Double
  Y As Double
  Z As Double
End Type
 
Public Function LENGTH(v as Vector3) As Double
  LENGTH = (v.x^2 + v.y^2 + v.z^2) ^ .5
End Function

[edit] Normalize

Normalizing a vector forms the basis of many more advanced vector operations. The process takes a vector of any length and preserves only its direction information (gives it a length of 1). This is useful when projecting one matrix into the axis of another with a dot product. Mathematically, a normalized vector is called a unit vector and can be obtained easily by dividing the vector by it's length: \hat v = \frac{\vec v}{\left | \vec v \right |}. In practice, however, each component (X, Y, and Z) of the vector is divided by it's length seperately.


[edit] C

typedef struct Vector3_
{
  double x;
  double y;
  double z;
} Vector3;
 
void Normalize(Vector3 *v)
{
  double len = length(v);
  v->x /= len;
  v->y /= len;
  v->z /= len;
}

[edit] C++

#include <cmath>
 
class Vector3
{
public:
	// Members
	double x, y, z;
 
	// Constructors
	Vector3() { } 
	Vector3( const Vector3 &r ) : x(r.x), y(r.y), z(r.z) { }
	Vector3( const double x, const double y, const double z ) : x(x), y(y),	z(z) { }
	~Vector3() { }
 
	// Operations
	Vector3 &operator = ( const Vector3 &r ) { x = r.x;	y = r.y; z = r.z; return *this;}
	Vector3 operator / ( const double r ) { return Vector3( x / r, y / r, z / r ); }
 
	// Methods
	double length() const {	return std::sqrt( x * x + y * y + z * z );	}
 
	Vector3 normalize() const
	{
		return *this / length();
	}
};

Note that there's a Complete math::vector Class elsewhere on the wiki that you can use instead of writing your own.

[edit] C#

public class Vector3
{
  public double X;
  public double Y;
  public double Z;
 
  public void Normalize()
  {
    // Vector3.Length property is under length section
    double length = this.Length;
 
    X /= length;
    Y /= length;
    Z /= length;
  }
}

[edit] Python

# v is a tuple representing a 3d vector
def normalize(v):
    len = length(v);
    return (v[0] / len, v[1] / len, v[2] / len)

[edit] Visual Basic

' In Public Object Module:
Public Type Vector3
  X As Double
  Y As Double
  Z As Double
End Type
 
Public Function NORMALIZE(ByRef v as Vector3)
  Dim VectorLen As Double
  VectorLen = LENGTH(v)
  v.X = v.X / VectorLen
  v.Y = v.Y / VectorLen
  v.Z = v.Z / VectorLen
End Function

[edit] The Dot Product

The dot product is very useful in game programming as it gives the angle between two vectors: \cos \theta = \frac{\vec v \cdot \vec w}{ \left | \vec v \right \vert \left | \vec w \right \vert} where θ is the angle between vectors \vec v and \vec w. On its own, the dot product is the length of the projection of \vec v onto the unit vector \hat w when the two vectors are placed so that their tails coincide.

[edit] C

typedef struct Vector3_
{
  double x;
  double y;
  double z;
} Vector3;
 
double dot(Vector3 *v, Vector3 *w)
{
  return (v->x*w->x + v->y*w->y + v->z*w->z);
}

[edit] C++

struct Vector3
{
  double x;
  double y;
  double z;
};
 
double dot(Vector3 const &v, Vector3 const &w)
{
  return (v.x*w.x + v.y*w.y + v.z*w.z);
}

Note that there's a Complete math::vector Class elsewhere on the wiki that you can use instead of writing your own.

[edit] C#

public class Vector3
{
  public double X;
  public double Y;
  public double Z;
 
  public static double Dot(Vector3 v, Vector3 w)
  {
    return (v.X*w.X + v.Y*w.Y + v.Z*w.Z);
  }
}

[edit] Python

# v and w are tuples representing 3d vectors
def dot(v, w):
    return v[0]*w[0] + v[1]*w[1] + v[2]*w[2]

[edit] Visual Basic

' In Public Object Module:
Public Type Vector3
  X As Double
  Y As Double
  Z As Double
End Type
 
Public Function DOT(v as Vector3, w as Vector3) As Double
  DOT = v.x*w.x + v.y*w.y + v.z*w.z
End Function

[edit] The Cross Product

The cross product is a mathematical operation that will return a resultant vector that is orthogonal to the two vectors used to calculate it and has a length relative to the lengths of the vectors and the angles between them: \left | \vec u \times \vec v \right | = \left | \vec u \right | \left | \vec v \right | \sin \theta

or (in terms of dot product):

\left | \vec u \times \vec v \right | = \left | \vec u \right | \left | \vec v \right | \sqrt{1-(\vec u \cdot \vec v)^2}

Visually, the orientation of a cross product resultant can be seen here:

Files:cross.gif


The cross product becomes very useful when calculating normals (vector orthogonal to a surface), which are used extensively in 3D game programming.

[edit] C

typedef struct Vector3_
{
  double x;
  double y;
  double z;
} Vector3;
 
Vector3 cross(Vector3 *v, Vector3 *w)
{
  Vector3 c = {
    v.y*w.z - v.z*w.y,
    v.z*w.x - v.x*w.z,
    v.x*w.y - v.y*w.x };
  return c;
}

[edit] C++

struct Vector3
{
  double x, y, z;
 
  Vector3(double set_x, double set_y, double set_z): 
  x(set_x), y(set_y), z(set_z) 
  {}
};
 
Vector3 cross(Vector3 const &v, Vector3 const &w)
{
  return Vector3(
    v.y*w.z - v.z*w.y,
    v.z*w.x - v.x*w.z,
    v.x*w.y - v.y*w.x );
}

Note that there's a Complete math::vector Class elsewhere on the wiki that you can use instead of writing your own.

[edit] C#

public class Vector3
{
  public double X;
  public double Y;
  public double Z;
 
  public static Vector3 Cross(Vector3 v, Vector3 w)
  {
    return new Vector3(
      v.Y*w.Z - v.Z*w.Y,
      v.Z*w.X - v.X*w.Z,
      v.X*w.Y - v.Y*w.X );
  }
}

[edit] Python

# v and w are tuples representing 3d vectors
def cross(v, w):
    x = v[1]*w[2] - v[2]*w[1]
    y = v[2]*w[0] - v[0]*w[2]
    z = v[0]*w[1] - v[1]*w[0]
 
    return (x, y, z)

[edit] Visual Basic

' In Public Object Module:
Public Type Vector3
  X As Double
  Y As Double
  Z As Double
End Type
 
Public Function CROSS(v as Vector3, w as Vector3) As Vector3
  CROSS.X = v.Y*w.Z - v.Z*w.Y
  CROSS.Y = v.Z*w.X - v.X*w.Z
  CROSS.Z = v.X*w.Y - v.Y*w.X
End Function

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