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Two Dimensional Space

Usually when we encounter mathematics in class we use two dimensional spaces. You can think of a two dimensional space as a plane that only has two directions.

 

line_directions

 

As you can see in the example above, no matter which way we turn on the point, we’re always going in only two directions. One common example of a two-dimensional space is a graph.

 

axes_on_an_axis
Here, we only have two directions: 

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.

 

two_dimensional
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.

 

line_equation
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.

line_equation_graph

 

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.

 

slope_intercept_equation
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.

 

slope_intercept_x_intercept_formula

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!

 

vector_direction
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:

 

three_dimensions
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_definition

 

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.

 

vector_graph
While vectors can also only have two components, x and y, vectors that are in three dimensions have three components. These components are always written in the following notation.

 

vector_notation

 

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.

 

vector_linear_equation
Here, we can break down what each element is. 

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.

vector_equation

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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Emma

Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.