::[ DISCLAIMER ]::
I will hold no responsibility to whatever happens to you, your computer, your sanity, your pet, or whatever that may happen to your existence for your reading the texts given in each tutorial. So in short, read at your own risk!

 
::[ INTRODUCTION ]::
Yes, wow, my first math tutorial! I must've bumped my head or somethin' to think of doing this (let's just say I was never one of the brightest person in school when it comes to math).

Anyway, I'm doing this mathematics tutorial to fill in the gap that I've found while searching for tutorials on graphics programming on the Internet. I hope this helps fill the gap!...:).

The first tutorial will be about vectors in the plane. "Vector" and "plane" are probably two of the most used words (and two of the most basic concepts needed) in the realm of three-dimensional graphics, that's why I'm introducing them here first.

SideNote: Everything that I will be discussing in my math tutorial series are based on what was taught to us (and how I understood the stuffs taught to us) at school.

 
::[ TUTORIAL ]::
So what is a vector? A vector is just a pair of real numbers written as <a,b> (for 2D vectors). If you have a 3D vector, you will write your vector as <a,b,c>...simple!

To elaborate further what a vector is, a vector is both a direction and magnitude/length. Let's say that (0,0) is the origin and you have a vector <1,1>, then the direction of your vector is from (0,0) towards <1,1>.

Here's an illustration:

              y-axis
                ^
                |
                +
                |
                |    
                + _ <1,1>
                | /|
                |/
      <-+---+---+---+----+--> x-axis
                |
                |
                v

From here onwards, whenever you see: "Vec A", this means I'm referring to a vector named A.

The next three paragraphs will be some definitions that you probably will never read again in any of my future math tutorials...but they are included for your knowledge and for completeness of the topic (you wouldn't want to be laughed at knowing all those advanced math stuffs without knowing the basics would you?...;) ). Don't worry, the definitions are easy...;).

• DEFINITION 1.1 The "a" in <a,b,c> is called the (drum rolls please) FIRST COMPONENT of the vector. The "b", on the other hand, is called the (stop drum rolls now please) SECOND COMPONENT of the vector. The "c" in...whazzat? You want to guess? Sure! What's your guess? THIRD COMPONENT? Wow! How did you know? You're a naturally born mathematician aren't you???...;).
• DEFINITION 1.2 Now, a GEOMETRIC REPRESENTATION of the 2D vector <a,b> is a directed line segment (ie, drawn like a ray) from any point (x,y) to the point (x+a,y+b).
• DEFINITION 1.3 The GEOMETRIC REPRESENTATION from (0,0) is called the POSITION REPRESENTATION of the vector.

Note that if I have <a,b> as my vector, then I have a 2D vector, else if I have <a,b,c>, then I have a 3D vector!

[ OPERATIONS ON VECTORS IN 3D-SPACE ]

Things get a bit interesting here. I will now introduce to you guys two operations that you can do with vectors on 3D-space (aka, R^3):

1. ADDITION: <a1,a2,a3> + <b1,b2,b3> = <a1+b1, a2+b2, a3+b3>
   EXAMPLE: <05,03,04> + <03,23,04> = <08,26,08>

2. SCALAR MULTIPLE: r<a1,a2,a3> = <ra1,ra2,ra3>
   EXAMPLE: 2<12,08,04> = <2(12),2(08),2(04)> = <24,16,08>

Obviously, operations on vectors in 3D-space is a no-brainer...:p.

And now, we laborously go through some properties resulting in my introduction about operations on vectors vectors in 3D-space... hehehe...=).

[ PROPERTIES OF ADDITION ]

1. Closure
   The sum of any two vector is a unique vector

   Commutativity

   Vec A + Vec B = Vec B + Vec A

   Associativity

   (Vec A + Vec B) + Vec C = Vec A + (Vec B + Vec C)

   Existence of Zero

   There exists a Vec O such that for any Vec A, Vec A + Vec O = Vec A

   For every Vec A, there exists a Vec -A such that Vec A + (Vec -A) = Vec O

[ PROPERTIES OF SCALAR MULTIPLE ]

1. Closure
   For any scalar r, and for any Vec A, r(Vec A) is a unique vector

   Distributive

   For any scalar r, and for any Vec A & Vec B, r(Vec A + Vec B) = r(Vec A) + r(Vec B)

   Distributive

   For any scalar r & scalar s, and for any Vec A, (r + s)Vec A = r(Vec A) + s(Vec A)

   For any scalar r, and scalar s, and for any Vec A, r(s(Vec A)) = (rs)Vec A

   Vec A(1) = Vec A

If you are one of those people who get easily intimidated by math because of all the symbols, well, don't be. If you look closer, there is nothing that I've written (or should I say "typed"? ;)) here that couldn't be understood by kids who have taken up standard multiplication and addition...;).

And now, more definitions (I know, I know, you're starting to get annoyed with them...but be forewarned, they will exist A LOT in my tutorial series...;) ).

• DEFINITION 1.4 The DIRECTION ANGLE of a nonzero 2D vector in the plane is the angle Q its POSITION REPRESENTATION forms with the positive x-axis, 0 <= Q <= 2(PI).

For example, we want to get the DIRECTION ANGLE of the vector <-2,2>:

              y-axis
                ^
                +
                |             -> tan Q = -2 / 2 = -1
                +             -> tan Q = -1
                |             -> Q = 3(pi) / 4
     <-2,2>  _  +  
            |\  |  
              \~+~_ 
               \|Q \
      <-+---+---+---+----+--> x-axis
                |
                +
                v

Guess what, more definitions...;)...:

• DEFINITION 1.5 If Vec A = <a1,a2>, we define its magnitude to be ||Vec A|| = square root(a1*a1 + a2*a2).

As an example to Definition 1.5:

    ||<-4,3>|| = square root(-4*-4 + 3*3) 
               = square root(16 + 9)
               = square root(25)
               = 5

• DEFINITION 1.6 A vector of magnitude / length equal to one is called the UNIT VECTOR.

It is very important to remember Definition 1.6 since you'll hear a lot about unit vectors when you are programming graphics (but if you're not into programming graphics...err...well, disregard this paragraph...:) ).

Example for Definition 1.6:

    Find a unit vector in the direction of Vec A = <5,-12>:

    Let's grab the magnitude first (using Definition 1.5)

      ||Vec A|| = square root(5*5 + (-12)*(-12))
                = square root(169)
                = 13

    Then the unit vector in the direction of Vec A Is:
   
                = <5/13, -12/13>

Simple, don't you think so?

• DEFINITION 1.7 The DIRECTION ANGLE of a nonzero 3D vector are the angles that its position representation forms with the positive x, positive y, and positive z axis, respectively named j, k, l. Note that the DIRECTION ANGLE is ALWAYS between 0 to pi.

Some formulas for Definition 1.7 (given Vec A = <a1,a2,a3>):

1. cos j = a1 / ||Vec A||
   cos j = a1 / square root(a1*a1 + a2*a2 + a3*a3)
2. cos k = a2 / ||Vec A||
3. cos l = a3 / ||Vec A||

cos j, cos k, and cos l are called the DIRECTION COSINES of the vector. You want to know the DIRECTION COSINES of the vector to, for example, know the angle j between a1 and ||Vec A||. Note that ||Vec A|| is the hypothenus...here's an illustration:

                 /|
                / |
               /  |
              /   |
             /    |
            /     |
           /      |
          /_     _|
         /j )   | |
        +-------+-+
        <--- a1 -->

Again, in the illustration above, ||Vec A|| is the hypothenus.

Guess what, MORE definitions!...

...kiddin'!...hehe...we're done actually...;).

 
::[ ENDING ]::
And that's that! Not hard eh? If there is anything that's not clear to you, your mind, or to that part of you that lets you comprehend the things here, you may email me at:

Well, that sums up my first math tutorial. I hope you guys like it! I'm sorry for the rather bad ASCII art...hehe...:/).

Part 2 will be about the dot product.

Briefly, the dot product accepts two vectors and the result is a single (*cough* scalar...;)) number. Its use and purpose you will know in Part 2!...:).  

This tutorial was written on: 12.14.2002