01) 2D Vector: Fundamentals

Where Vector means "determining the position of one point in space relative to another".

Things to note:

1) While not always the case, Point O is usually considered the origin point, meaning that it's position in 2 dimensional space is (0,0).

2) Due to the limitations of digital text, any Vector going by the naming convention of XY where X and Y are points, implies a Vector Line from Point X to Point Y. Where it would usually look, for OA, like:



3) Vector Lines can be added together, the rule being that the end point of one must be the beginning point of the next, for example: OA + AB = OB, since OA is the vector from Point O to Point A, then AB is the vector from Point A to Point B, adding them together creates a vector that goes from Point O to Point A to Point B.

4) Vectors can also be denoted in a Variable-like fashion, for example OA can be denoted as "a". Thus these Vectors can be arranged in an equation to form the Vector of one point to another.

For example, if OA = "a", and OB = "b", and we wish to find AB then we must find a way to go from Point B to Point A. OA is a vector from Point O to Point A, however if it were heading the other way, from Point A to point O then it would be represented as AO and as "-a" due to the Vector going the opposite direction of "a" with the same length. Thus to get AB we can add AO to OB represented as "-a + b" simplified as "b - a", to get a Vector that goes from Point A to Point B.

5) Vectors can be denoted in a Matrix, such as:


Where 3 is the x-coordinate, and 4 is the y-coordinate. Vectors can be multiplied and divided by any real or complex number, thereby scaling by that amount.

6) The Magnitude of a Vector is the length of a vector, since the x-coordinate and y-coordinate of a vector are perpendicular, the Pythagorean Theorem can be used, where the length is found by:



7) A Unit Vector is a Vector based off another vector, where the magnitude is equal 1 but the direction remains the same. The easiest method of finding the Unit Vector is by dividing the original Vector by it's magnitude. A Unit Vector can be shown in either way:



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