In this explainer, we will learn how to represent a vector in space using a three-dimensional coordinate system.
We will find the components of a vector that connects two points in three-dimensional space using known coordinates, as well as the addition and subtraction of vectors. We will then find the coordinates of an unknown point using the coordinates of a known point and the components of the known vector between them. Finally, we will find the components of a three-dimensional vector that is represented graphically.
The direction of a vector represented graphically is denoted by an arrow. It has a tail (initial point) and head (terminal point). Let us begin by considering what we mean by a unit vector.
Definition: Unit Vectors
A unit vector is a vector of magnitude equal to 1. The unit vectors in the -, -, and -directions are denoted by , , and respectively.
Any vector can be written in the form . These can be alternatively represented as and .
In our first example, we will consider the unit vector in the direction of one of the axes.
Example 1: Finding the Unit Vector in the Direction of the 𝑦-axis
Find the unit vector in the direction of the -axis.
Let us consider the three-dimensional coordinate grid, with origin . Any vector can be written in the form , where , , and are the components of the vector in each of those directions. The unit vectors in each of these positive directions are denoted by , , and respectively.
Unit vectors have magnitude 1, and we are told in the question that the unit vector moves in the direction of the -axis. We can choose to start at the origin, which has coordinates , as shown in the following diagram.
We can see in our diagram that the vector travels entirely in the -direction, so its - and -components must be equal to zero. If and , then, in order for the vector to have a positive direction and a magnitude (length) of 1, the -component must be equal to 1.
The unit vector, , in the direction of the -axis is equal to .
Similarly, the unit vector in the direction of the -axis is equal to and the unit vector in the direction of the -axis is equal to .
We will now consider any vector in space that starts at the origin.
In the diagram below, point has coordinates and vector begins at the origin and ends at point .
From the origin, we move 2 units in the positive -direction, 5 units in the positive -direction, and 3 units in the positive -direction; so vector .
Definition: Vector from the Origin
If point has coordinates , then where the components , , and are the displacements of point in the -, -, and -directions from the origin.
The direction of the vector is important, as vector starts at the origin and finishes at point , whereas vector would start at point and finish at the origin. This means that the vectors would have opposite directions, and so each of the components will have the opposite sign.
If , then .
Definition: Vector in the Opposite Direction
For any point ,
This means that and if , then, .
Let us now consider how to find an expression for a vector in three-dimensional space between any two points.
Example 2: Understanding Vectors between Two Given Points
Which of the following is equal to the vector ?
Let us begin by considering two distinct points and in space, as shown in the diagram.
We want to construct the vector from to , which is denoted :
In order to do this, we can travel via the origin as shown in the diagram below. We can go from point to point , and then from point to point :
This can be written as the following equation using vectors:
For any point , .
We can use this property to rewrite our equation as follows:
Any vector from the origin to a given point will have -, -, and -components equal to the coordinates of the point, so and .
We can use these to rewrite our equation:
Therefore, the correct answer is option C.
Point and vector have the same components, but it is important to note that one of them is a point and the other is a vector. We generalize the idea that we can find vector by subtracting vector from vector .
This leads us to the following rule when dealing with the vector between two points.
Definition: Vector between Two Points
The vector between any two points and is given by
The direction of the vector is important. We know that and have opposite directions and the same magnitude. When we add these vectors together, the displacement is 0; this gives us a useful result.
Definition: Vector in the Opposite Direction
For any points in space, and ,
This means that
In our next example, we will find the vector between two given points in space.
Example 3: Finding the Vector between Two Given Points
Given and , find .
For any two points and , we generalize the idea that we can find vector by subtracting vector from vector :
Hence, the vector from to is given by .
We will now consider the general equation for a position vector.
Definition: Equation for Position Vector
For any points and , together with the origin ,
We can add to both sides of this equation to get
In the fourth example, we find the position vector of an unknown point using the position vector of a known point and the components of the vector between the points.
Example 4: Finding the Position Vector of a Point given the Vector Joining It to Another Given Point
Given and , express in terms of the fundamental unit vectors.
Recall that, for any two points and , we generalize the idea that we can find vector by subtracting vector from vector :
Adding to both sides of the equation gives us
To add vectors together, we add the corresponding components.
Substituting the components of and into our equation gives us
In our final two examples, we will find the components of a three-dimensional vector represented graphically.
Example 5: Finding the Components of a 3D Position Vector That Is Represented Graphically
Using the graph, write the vector in terms of its components.
We can see from the figure that point has coordinates , as we move 2 units in the positive -direction, 3 units in the positive -direction, and 4 units in the positive -direction.
Vector begins at the origin and ends at point . This means that . The vector will have -, -, and -components equal to the coordinates of point :
Example 6: Finding the Components of a 3D Vector That Is Represented Graphically
Find the vector using the graph.
We recall that the vector between any two points and is given by
From the diagram, we see that vertex has coordinates and vertex has coordinates .
Alternatively, we can solve this problem using position vectors. We know that vector can be calculated by subtracting vector or from vector or :
Vertex has coordinates , which means that . Likewise, vertex has coordinates , which means that .
Subtracting the corresponding components gives us
As the given figure is a cube of side length 3, we know that each of the components must be equal to 3, which supports the answer that vector .
We will finish this explainer by recapping some of the key points.
- A vector in 3D space can be written in component form, , or in terms of the fundamental unit vectors, .
- The vector connecting a point , in space, from the origin, , can be written as or and will have components equal to the coordinates of point .
- The vector connecting two points and , written , can be calculated by subtracting vector from vector such that .