Table Of Contents

• Introduction
• Addition and Substraction and Scaling
• Scalar Product
• Cross Product
• Length of a Vector
• Examples of Vector math
  • Distance between two Points
  • Plane from 3 Points
  • Distance Point to Plane
• Final Words

• Introduction

  This is a short primer on vector mathematics. I know that there are lots of vector tutorials out there.
  This text is not intendet as a replacement for a good math-book, but it might be a good reference.
  
  Vector math is important for all kinds of 3d programming. And since this pages are more or less 3d-related I only deal with 3 dimensional vectors.
  
  Since most browsers still don't support greek and unicode characters it's a little bit tricky to put them into the text.. I hope that your browser translates them correctly. My netscape refuses to do so.
  
  

  How I write my equations:
  
  

  1. Vectors are notated in lower case italic chars (I usually use a, b, c and so on).
  2. Scalars are notated in lower case chars (non-italic). By the way, if you don't know what a scalar value is: they are the normal, regular number like 24.2 you've ever calculated with before.

  
  
  

• Addition and Substraction and Scaling

  Addition and Substraction is done on each component on the vector:
  
  Addition:

       c   = a   + b
  
  
       c.x = a.x + b.x
  
       c.y = a.y + b.y
  
       c.z = a.z + b.z
  
  

  
  Subtraction:

       c   = a   - b
  
  
       c.x = a.x - b.x
  
       c.y = a.y - b.y
  
       c.z = a.z - b.z
  
  

  Scaling (Division and Multiplication with a scalar)

       c   = a   * b
  
  
       c.x = a.x * b
  
       c.y = a.y * b
  
       c.z = a.z * b
  
  

  
  
  

• Scalar Product

  
  
  The scalar product of two vectors a and b is calculated this way:

       c = a.x*b.x + a.y*b.y + a.z*b.z

  It has two different meanings:
  
  
  1. c is the projection of vector a onto b (or vice versa).
  2. c = cos(β) the cosine of the angle between the vectors a and b.
  
  The scalar product is also called inner product and dot-product.
  Its usually notated as a point c = a . b
  
  

• The Cross Product

  
  
  The cross product of two vectors a and b is calculated this way:

       c.x = a.y*b.z - a.z*b.y
  
       c.y = a.x*b.z - a.z*b.y
  
       c.z = a.x*b.y - a.y*b.z
  
  

  Again the cross-product has several properties:
  
  
  1. The length of c = A which equals to the area of the parallelogram formed by a and b
  2. c stands perpendicular on the plane formed by a and b.
     
     In other words: (a . c) = (b . c) = 0
     
  
  The cross product is also called outer product or vector product
  Its usually notated as a x or star (I use the x): c = a x b
  
  
  

• Length of a Vector

  The euclidean length of a vector (we normally only deal with the euclidean length, there are other definitions, but they're not that important) is defined as:

       |v| = square_root(v.x² + v.y² + v.z²)
  
  

  Sometimes you hear about a normalized vector. That means, you scale a vector this way, that it's length becomes one.

       v'.x = v.x / |v|
  
       v'.y = v.y / |v|
  
       v'.z = v.z / |v|
  
  

  Of cause this only works if the vector is no null-vector (|v| = 0).
  I think you already guessed that the two bars |v| are used to notate the length.
  
  
  

• Examples of Vector Math

  Here are some examples..
  
  

  1. Distance between two points
     
     We start simple: What's the distance between two points
     

     
     
           a = (10,20,30)
     
           b = (11,12,13)
     
         

     build a third vector c from a to b:

     
     
           c = b - a
     
           c.x = b.x - a.x
     
           c.y = b.y - a.y
     
           c.z = b.z - a.z
     
         

     Calculate the length of c:

     
     
           distance = |c|
     
           distance = square_root(c.x² + c.y² + c.z²)
     
         

     
     
  2. Plane from 3 Points
     
     A plane can be expressed in several ways.
     
     One way to express a plane is to give the position of three Points on the plane. Another way is to define it by the normal-vector which is perpendicular to the plane (see cross-product), and the distance from the plane to the origin.
     
     It's very usefull if you want to make distance or intersection calculation with other planes, lines or points.
     
     Given are three points a,b and c
     
     Build the two vectors:

            u = c-a
     
            v = c-b
     
         

     The vector perpedicular to u and v is:

            n = u x v
     
         

     Get the normalized vector n' = n / |n|
     build the distance d:
     

            d = n' . a
     
         

     You can also use the points b or c. It doesn't matter. It's only important that the point you dot the normal vector with lies on the plane.
     
     Here comes the numerical example:

     
     
     
     
                | 1 |         | 2 |         | 1 |
     
           a =  | 2 |     b = | 3 |     c = | 0 |
     
                | 3 |         | 4 |         | 3 |
     
     
     
     
     
                | 1-1 |   |  0 |          | 1-2 |   | -1 |
     
           u =  | 0-2 | = | -2 |      v = | 0-3 | = | -3 |
     
                | 3-3 |   |  0 |          | 3-4 |   | -1 |
     
     
     
     
     
                |  (-2 * -1) - ( 0 * -3) |     |  2 - 0 |  =  |  2 |
     
           n =  |  ( 0 * -1) - (-1 *  0) |  =  |  0 - 0 |  =  |  0 |
     
                |  (-2 * -1) - (-3 *  0) |     | -2 - 0 |  =  | -2 |
     
     
     
     
     
     
     
           |n|  = sqrt (8)
     
     
     
                            |  0,707 |
     
           n'   = n/|n|  =  |  0.000 |
     
                            | -0,707 |
     
     
     
           d    = n' . a  = ( 0.707 - 2,121) = -1.414
     
     
     
           (try dotting with another point (b or c). you sould get -1.414 in all cases)
     
     
     
           Equation of the plane is:     p . n' - d = 0
     
     
     
           All points p that solve this equations lie on the plane.
     
         

  3. Distance Point to Plane
     
     Allright, how can we use this equation for something usefull?
     
     We can calculate the distance from the point to a plane. Given the is the plane equation p . n' -d = 0 How do we get the distance from a point to that plane? Easy! Just let p be the point you want to work with and evaluate the equation
     

            distance = p . n' -d
     
         

     This formula has a really great application in 3d graphics. If you already read my Backface-Culling in Object-Space tutorial you might find this equation familiar. To be exact: it's the same equation.
     
     If the distance of the camera to a polygon-plane is <, then that the normal-vector of the plane is facing away from the camera.
     
     Since we usually work with one-sided polygons in the 3d-world this is a cheap way to do backface culling. (if you don't know. you usually only draw polygons with normal-vector facing to the camera. This saves lots of unnessesary calculations).

• Final Words

  As long as I didn't knew what to do with the math I learn at school or in the university, it was just boring and useless stuff.
  
  But as soon as you can use it, math can be fun.

  If you want to contact me you might write me a mail.

  And now bring me back to the Code-Corner!