Two Dimensional Space
| Direction 1 | X axis |
| Direction 2 | Y axis |
In two dimensional spaces like a graph, we can have any set of points. These points have coordinates, which are like directions that we can use to find where that point is. In two dimensional words, we can only go in two directions: left or right, up or down, etc.
| A | X axis | Go to the left or right |
| B | Y axis | Go up or down |
Definition of a Line
You’ve probably encountered lines both inside and outside of maths class. While we tend to think about lines as solid, straight elements - you can also think of a line as a set of points.
| Point A | (1,2) |
| Point B | (5,8) |
If you extend this notion, you can also think about a line as a set of many different points. When we look at this in two dimensional space, we can actually start to predict the behaviour of a line based on the information we have about all those points.
Equation of a Line
The behaviour of a line is essentially the direction that line goes in. When we have a set of points, we can actually get the equation of the line that runs through them. Take a look at the most common notation for the equation of a line.
| y | Y coordinate |
| m | Slope of the line; you can think of this as the direction the line goes in |
| x | X coordinate |
| b | Y intercept; the point where the line would touch, or intercept, the y axis |
When we have the equation of a line, we can get the y coordinate from the x coordinate and vice versa. Take a look at the example below.
| y | 0 |
| m | 2 |
| x | ? |
| b | 4 |
In order to get the x coordinate, we would simply rearrange the formula to solve for x.
Three Dimensional Space
When we talk about vectors, we’re essentially still talking about lines. However, these lines can exist in three dimensional space. What exactly is three dimensional space though? Well, it’s everything around you!
| A | Left or right |
| B | Up or down |
| C | In or out |
As you can see, we can go in three directions in three dimensional space. The easiest way to understand how our lives are lived in three dimensions is to think about a movie. When you see a movie, you’re watching the characters on a plane.
If the characters were able to come out of the screen, they would go either out or in:
| Direction #1 | Left |
| Direction #2 | Up |
| Direction #3 | Out |
Higher Order Spaces
Vectors don’t just exist in three dimensional spaces, but also in higher order spaces. Another word for the order of something is the degree. Take for example any number to the power n.
| Power | Order |
| n = 1 | 1 |
| n = 2 | 2 |
| n = 10 | 10 |
The order, power or degree indicates the quantity of powers. So, you can have the following:
| Order = 4 | 4 dimensional space |
| Order = 5 | 5 dimensional space |
| Order = 6 | 6 dimensional space |
Vector Definition
A vector is a line in three dimensional space. You can think of vectors as lines that have a magnitude and a direction.
| Magnitude | Distance between starting point and ending point |
| Direction | The direction that magnitude goes in |
When we have a vector, we can go in three directions. Take a look below.
 | The vector |
| Array (1 2 5) | The vector array |
Vector Equation of a Line
When we’re working in two dimensions, recall that the formula of a line is simply y = mx + b. However, when we have a vector in three dimensions, things can get a little more complicated. Let’s get an overview of the vector equation of a line in three dimensions.
| Element | Description |
 | The vector that contains point P |
 | The straight line that passes through a point  with vector |
| t | The value that multiplies vector |
| The vector that is parallel to the one that contains point P |
Let’s take a look at an example point.
Say that point P is (3,0,1), which is located on the vector . The vector that is parallel to vector has the following equation: -2
+ 
- 
. In order to find the equation for vector , we simply plug these points in.
\[
\vec{r} = 3\vec{i} - \vec{k} + t(-2\vec{i} + \vec{j} - \vec{k})
\]
Keep in mind that point P has the following form.
| Point P | Standard Form |
| (3,0,1) |  |
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