Vector

A vector has magnitude and direction. In a 2D plane, this can be represented by a coordinate points. Mathematic notation uses bold or an arrow over the variable to indicate a $\overrightarrow{vector}$ and vertical bars to represent $||\overrightarrow{magnitude}||$ of a vector.

Magnitude can be calculated as: $||\vec{z}||=\sqrt{{\vec{x}}^2+{\vec{y}}^2}$

Example

$$\begin{array}{rl}||\vec{z}||& =\sqrt{4^2+3^2}\\ & =\sqrt{25}\\ & =5\end{array}$$

Normalized Vector

A Normalized vector is a vector with the same direction but a magnitude of 1. It is calculated as: $Normal(x)=\frac{x}{||x||}$

Mathematical Operations

Basic operations can be applied to vectors in order to create new vectors. In all examples, assume the following:

$\vec{a}=(2,5)$ and $\vec{b}=(6,7)$

Addition

Vector addition adds corresponding indices with each other.

$(x_1,y_1)+(x_2,y_2)=(x_1+x_2,y_1+y_2)$

Example

$$\begin{array}{rl}\vec{c}& =\vec{a}+\vec{b}\\ & =(2,5)+(6,7)\\ & =(2+6,5+7)\\ & =(8,12)\end{array}$$

Subtraction

Vector subtraction involves reversing the second vector and then performing addition.

$(x_1,y_1)-(x_2,y_2)=(x_1,y_1)+(-x_2,-y_2)$

Example

$$\begin{array}{rl}\vec{c}& =(2,5)-(6,7)\\ & =(2,5)+(-6,-7)\\ & =(2-6,5-7)\\ & =(-4,-2)\end{array}$$

Scaling

Scaling a vector can be achieved by multiplying the vector with a Scalar.

$(x_1,y_1)\ast ||\vec{z}||=(x_1\ast ||\vec{z}||,y_1\ast ||\vec{z}||)$

Example

$$\begin{array}{rl}||\vec{z}||& =3\\ \vec{c}& =\vec{a}\ast ||\vec{z}||\\ & =(2,5)\ast 3\\ & =(2\ast 3,5\ast 3)\\ & =(6,15)\end{array}$$

Dot Product

The Dot Product is a method of multiplying two vectors to return a scalar. It can be used to calculate a reflection angle in 2D Space.

Assume $\theta$ as the angle between vectors $\vec{a}$ and $\vec{b}$

$\vec{a}\cdot \vec{b}=||\vec{a}||\ast ||\vec{b}||\ast cos(\theta )$