Lesson Explainer: Vectors in Space Mathematics

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 ๐‘ฅ-, ๐‘ฆ-, and ๐‘ง-directions are denoted by โƒ‘๐‘–, โƒ‘๐‘—, and โƒ‘๐‘˜ respectively.

Any vector can be written in the form ๐‘ฅโƒ‘๐‘–+๐‘ฆโƒ‘๐‘—+๐‘งโƒ‘๐‘˜. These can be alternatively represented as (๐‘ฅ,๐‘ฆ,๐‘ง) and ๏€ฟ๐‘ฅ๐‘ฆ๐‘ง๏‹.

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 ๐‘ฆ-axis

Find the unit vector in the direction of the ๐‘ฆ-axis.

Answer

Let us consider the three-dimensional coordinate grid, with origin ๐‘‚. Any vector can be written in the form (๐‘ฅ,๐‘ฆ,๐‘ง), where ๐‘ฅ, ๐‘ฆ, and ๐‘ง are the components of the vector in each of those directions. The unit vectors in each of these positive directions are denoted by โƒ‘๐‘–, โƒ‘๐‘—, and โƒ‘๐‘˜ respectively.

Unit vectors have magnitude 1, and we are told in the question that the unit vector moves in the direction of the ๐‘ฆ-axis. We can choose to start at the origin, which has coordinates (0,0,0), as shown in the following diagram.

We can see in our diagram that the vector travels entirely in the ๐‘ฆ-direction, so its ๐‘ฅ- and ๐‘ง-components must be equal to zero. If ๐‘ฅ=0 and ๐‘ง=0, then, in order for the vector to have a positive direction and a magnitude (length) of 1, the ๐‘ฆ-component must be equal to 1.

The unit vector, โƒ‘๐‘—, in the direction of the ๐‘ฆ-axis is equal to (0,1,0).

Similarly, the unit vector in the direction of the ๐‘ฅ-axis is equal to (1,0,0) and the unit vector in the direction of the ๐‘ง-axis is equal to (0,0,1).

We will now consider any vector in space that starts at the origin.

In the diagram below, point ๐ด has coordinates (2,5,3) and vector ๏ƒ ๐‘‚๐ด begins at the origin and ends at point ๐ด.

From the origin, we move 2 units in the positive ๐‘ฅ-direction, 5 units in the positive ๐‘ฆ-direction, and 3 units in the positive ๐‘ง-direction; so vector ๏ƒ ๐‘‚๐ด=(2,5,3).

Definition: Vector from the Origin

If point ๐ด has coordinates (๐‘ฅ,๐‘ฆ,๐‘ง), then ๏ƒ ๐‘‚๐ด=(๐‘ฅ,๐‘ฆ,๐‘ง), where the components ๐‘ฅ, ๐‘ฆ, and ๐‘ง are the displacements of point ๐ด in the ๐‘ฅ-, ๐‘ฆ-, and ๐‘ง-directions from the origin.

The direction of the vector is important, as vector ๏ƒ ๐‘‚๐ด starts at the origin and finishes at point ๐ด, whereas vector ๏ƒ ๐ด๐‘‚ would start at point ๐ด 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 ๏ƒ ๐‘‚๐ด=(2,3,5), then ๏ƒ ๐ด๐‘‚=(โˆ’2,โˆ’3,โˆ’5).

Definition: Vector in the Opposite Direction

For any point ๐ด, ๏ƒ ๐‘‚๐ด+๏ƒ ๐ด๐‘‚=โƒ‘๐‘‚.

This means that ๏ƒ ๐‘‚๐ด=โˆ’๏ƒ ๐ด๐‘‚, and if ๏ƒ ๐‘‚๐ด=(๐‘ฅ,๐‘ฆ,๐‘ง), then, ๏ƒ ๐ด๐‘‚=(โˆ’๐‘ฅ,โˆ’๐‘ฆ,โˆ’๐‘ง).

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 ๏ƒ ๐ด๐ต?

  1. โƒ‘๐ดโˆ’โƒ‘๐ต
  2. โƒ‘๐ด+โƒ‘๐ต
  3. โƒ‘๐ตโˆ’โƒ‘๐ด
  4. โƒ‘๐ดร—โƒ‘๐ต

Answer

Let us begin by considering two distinct points ๐ด and ๐ต in space, as shown in the diagram.

We want to construct the vector from ๐ด to ๐ต, which is denoted ๏ƒ ๐ด๐ต:

In order to do this, we can travel via the origin as shown in the diagram below. We can go from point ๐ด to point ๐‘‚, and then from point ๐‘‚ to point ๐ต:

This can be written as the following equation using vectors: ๏ƒ ๐ด๐ต=๏ƒ ๐ด๐‘‚+๏ƒŸ๐‘‚๐ต.

For any point ๐ด, ๏ƒ ๐ด๐‘‚=โˆ’๏ƒ ๐‘‚๐ด.

We can use this property to rewrite our equation as follows: ๏ƒ ๐ด๐ต=โˆ’๏ƒ ๐‘‚๐ด+๏ƒŸ๐‘‚๐ต.

Any vector from the origin to a given point will have ๐‘ฅ-, ๐‘ฆ-, and ๐‘ง-components equal to the coordinates of the point, so ๏ƒ ๐‘‚๐ด=โƒ‘๐ด and ๏ƒŸ๐‘‚๐ต=โƒ‘๐ต.

We can use these to rewrite our equation: ๏ƒ ๐ด๐ต=โˆ’๏ƒ ๐‘‚๐ด+๏ƒŸ๐‘‚๐ต=โˆ’โƒ‘๐ด+โƒ‘๐ต=โƒ‘๐ตโˆ’โƒ‘๐ด.

Therefore, the correct answer is option C.

Point ๐ด and vector ๏ƒ ๐‘‚๐ด 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 ๏ƒ ๐ด๐ต by subtracting vector โƒ‘๐ด๏€บ๏ƒ ๐‘‚๐ด๏† from vector โƒ‘๐ต๏€บ๏ƒŸ๐‘‚๐ต๏†.

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 ๐ด(๐‘ฅ,๐‘ฆ,๐‘ง)๏Šง๏Šง๏Šง and ๐ต(๐‘ฅ,๐‘ฆ,๐‘ง)๏Šจ๏Šจ๏Šจ is given by ๏ƒ ๐ด๐ต=(๐‘ฅโˆ’๐‘ฅ,๐‘ฆโˆ’๐‘ฆ,๐‘งโˆ’๐‘ง).๏Šจ๏Šง๏Šจ๏Šง๏Šจ๏Šง

The direction of the vector is important. We know that ๏ƒ ๐ด๐ต and ๏ƒ ๐ต๐ด 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, ๐ด and ๐ต, ๏ƒ ๐ด๐ต+๏ƒ ๐ต๐ด=(0,0,0).

This means that ๏ƒ ๐ด๐ต=โˆ’๏ƒ ๐ต๐ด.

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 โƒ‘๐ด=(6,1,4) and โƒ‘๐ต=(3,1,2), find ๏ƒ ๐ด๐ต.

Answer

For any two points ๐ด and ๐ต, we generalize the idea that we can find vector ๏ƒ ๐ด๐ต by subtracting vector โƒ‘๐ด๏€บ๏ƒ ๐‘‚๐ด๏† from vector โƒ‘๐ต๏€บ๏ƒŸ๐‘‚๐ต๏†: ๏ƒ ๐ด๐ต=โƒ‘๐ตโˆ’โƒ‘๐ด.

This means ๏ƒ ๐ด๐ต=(3,1,2)โˆ’(6,1,4)=(3โˆ’6,1โˆ’1,2โˆ’4)=(โˆ’3,0,โˆ’2).

Hence, the vector from ๐ด to ๐ต is given by (โˆ’3,0,โˆ’2).

We will now consider the general equation for a position vector.

Definition: Equation for Position Vector

For any points ๐ด and ๐ต, together with the origin ๐‘‚, ๏ƒ ๐ด๐ต=๏ƒŸ๐‘‚๐ตโˆ’๏ƒ ๐‘‚๐ด.

We can add ๏ƒ ๐‘‚๐ด to both sides of this equation to get ๏ƒ ๐ด๐ต+๏ƒ ๐‘‚๐ด=๏ƒŸ๐‘‚๐ต.

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 ๏ƒ ๐ด๐ต=(โˆ’1,โˆ’3,0) and โƒ‘๐ด=(โˆ’4,โˆ’5,โˆ’5), express โƒ‘๐ต in terms of the fundamental unit vectors.

Answer

Recall that, for any two points ๐ด and ๐ต, we generalize the idea that we can find vector ๏ƒ ๐ด๐ต by subtracting vector โƒ‘๐ด๏€บ๏ƒ ๐‘‚๐ด๏† from vector โƒ‘๐ต๏€บ๏ƒŸ๐‘‚๐ต๏†: ๏ƒ ๐ด๐ต=โƒ‘๐ตโˆ’โƒ‘๐ด.

Adding โƒ‘๐ด to both sides of the equation gives us ๏ƒ ๐ด๐ต+โƒ‘๐ด=โƒ‘๐ต.

To add vectors together, we add the corresponding components.

Substituting the components of โƒ‘๐ด and ๏ƒ ๐ด๐ต into our equation gives us โƒ‘๐ต=(โˆ’1,โˆ’3,0)+(โˆ’4,โˆ’5,โˆ’5)=(โˆ’1+(โˆ’4),โˆ’3+(โˆ’5),0+(โˆ’5))=(โˆ’5,โˆ’8,โˆ’5).

Therefore, โƒ‘๐ต=โˆ’5โƒ‘๐‘–โˆ’8โƒ‘๐‘—โˆ’5โƒ‘๐‘˜.

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 โƒ‘๐ด in terms of its components.

Answer

We can see from the figure that point ๐ด has coordinates (2,3,4), as we move 2 units in the positive ๐‘ฅ-direction, 3 units in the positive ๐‘ฆ-direction, and 4 units in the positive ๐‘ง-direction.

Vector โƒ‘๐ด begins at the origin and ends at point ๐ด. This means that โƒ‘๐ด=๏ƒ ๐‘‚๐ด. The vector will have ๐‘ฅ-, ๐‘ฆ-, and ๐‘ง-components equal to the coordinates of point ๐ด: โƒ‘๐ด=(2,3,4).

Example 6: Finding the Components of a 3D Vector That Is Represented Graphically

Find the vector ๏ƒ ๐ด๐บ using the graph.

Answer

We recall that the vector between any two points ๐ด(๐‘ฅ,๐‘ฆ,๐‘ง)๏Šง๏Šง๏Šง and ๐ต(๐‘ฅ,๐‘ฆ,๐‘ง)๏Šจ๏Šจ๏Šจ is given by ๏ƒ ๐ด๐ต=(๐‘ฅโˆ’๐‘ฅ,๐‘ฆโˆ’๐‘ฆ,๐‘งโˆ’๐‘ง).๏Šจ๏Šง๏Šจ๏Šง๏Šจ๏Šง

From the diagram, we see that vertex ๐ด has coordinates (1,1,0) and vertex ๐บ has coordinates (4,4,3).

Therefore, ๏ƒ ๐ด๐บ=(4โˆ’1,4โˆ’1,3โˆ’0)๏ƒ ๐ด๐บ=(3,3,3).

Alternatively, we can solve this problem using position vectors. We know that vector ๏ƒ ๐ด๐บ can be calculated by subtracting vector โƒ‘๐ด or ๏ƒ ๐‘‚๐ด from vector โƒ‘๐บ or ๏ƒ ๐‘‚๐บ: ๏ƒ ๐ด๐บ=๏ƒ ๐‘‚๐บโˆ’๏ƒ ๐‘‚๐ด.

Vertex ๐ด has coordinates (1,1,0), which means that ๏ƒ ๐‘‚๐ด=(1,1,0). Likewise, vertex ๐บ has coordinates (4,4,3), which means that ๏ƒ ๐‘‚๐บ=(4,4,3).

Therefore, ๏ƒ ๐ด๐บ=(4,4,3)โˆ’(1,1,0).

Subtracting the corresponding components gives us ๏ƒ ๐ด๐บ=(4โˆ’1,4โˆ’1,3โˆ’0)=(3,3,3).

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 ๏ƒ ๐ด๐บ=(3,3,3).

We will finish this explainer by recapping some of the key points.

Key Points

  • A vector in 3D space can be written in component form, (๐‘ฅ,๐‘ฆ,๐‘ง), or in terms of the fundamental unit vectors, ๐‘ฅโƒ‘๐‘–+๐‘ฆโƒ‘๐‘—+๐‘งโƒ‘๐‘˜.
  • The vector connecting a point ๐ด, in space, from the origin, ๐‘‚, can be written as โƒ‘๐ด or ๏ƒ ๐‘‚๐ด and will have components equal to the coordinates of point ๐ด.
  • The vector connecting two points ๐ด and ๐ต, written ๏ƒ ๐ด๐ต, can be calculated by subtracting vector โƒ‘๐ด from vector โƒ‘๐ต such that ๏ƒ ๐ด๐ต=๏ƒŸ๐‘‚๐ตโˆ’๏ƒ ๐‘‚๐ด.

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