# Lesson Explainer: Points, Midpoints, and Distances in Space Mathematics

In this explainer, we will learn how to find the coordinates of a point in 3D, the distance between two points in 3D, and the coordinates of a midpoint and an endpoint in 3D using the formula.

We should already be aware of how to find all of these in two dimensions. Any point in two dimensions will have an - and -coordinate and can be written in the form . Each of the real numbers in the ordered pair represents the displacement of that point from the origin, in other words, the positive or negative distance from the point .

If two points and have coordinates and , respectively, then we can calculate their midpoint by using the formula .

If two points and have coordinates and , respectively, then we can calculate the distance between them by using the distance formula, derived from the Pythagorean theorem, .

We will investigate in this explainer how we can extend these formulae to include a third coordinate when dealing with points in three dimensions.

### Definition: Coordinates of a Point in 3D Space

Any point in three dimensions will have -, -, and -coordinates and can be written in the form . Each of the real numbers in the ordered triple gives the distance from the origin measured along the corresponding axis.

In our first example, we will consider in which plane a point, with one of its coordinates equal to zero, lies.

### Example 1: Identifying the Plane in Which a Given Coordinate Lies

In which of the following coordinate planes does the point lie?

We know that a point in 3D will have -, -, and -coordinates. In this question, , , and .

As the -coordinate is equal to zero, the point lies a distance of zero from the origin in the -direction. This means that it will lie on the -plane. In fact, any point with coordinates will lie on this plane.

We can therefore conclude that the point lies on the -plane.

### Definition: The Three Coordinate Planes

Any point with coordinates will lie on the -plane.

Similarly, any point with coordinates will lie on the -plane, and any point with coordinates will lie on the -plane.

In our next question, we will consider how we can work out the coordinates of a point in three dimensions.

### Example 2: Finding the Coordinates of a Given Point in 3D

Determine the coordinates of point .

Any point in three dimensions will have -, -, and -coordinates and can be written in the form .

Moving from the origin, we travel 3 units in the positive -direction, units in the -direction, and finally 3 units in the -direction.

This means that , , and .

The coordinates of point are .

We recall that the midpoint formula in two dimensions is simply telling us to find the average value of two points. We find the average of the -coordinates and the average of the -coordinates. We will now extend this idea to three dimensions by finding the average of the -coordinates as well.

To find the average of any two numbers, we add them and then divide their sum by two.

### Definition: The Midpoint of Two Points in 3D Space

If two points and have coordinates and , respectively, then we can calculate their midpoint by using the following formula:

In our next example, we will use this formula to work out the midpoint of two points in space.

### Example 3: Finding the Coordinates of a Midpoint in 3D

Points and have coordinates and respectively. Determine the coordinates of the midpoint of .

In order to find the midpoint of two points in three dimensions, we will use the formula to calculate the midpoint of coordinates and :

We will let point have coordinates and point have coordinates .

The midpoint between points and is

The coordinates of the midpoint of are .

In our next example, we will use the midpoint formula to work out an endpoint given the midpoint of two points in space and another endpoint.

### Example 4: Finding the Coordinates of an Endpoint of a Line Segment given the Coordinates of the Midpoint and the Coordinates of a Start Point

Given that point is the midpoint of and that , what are the coordinates of ?

In order to find the midpoint of two points in three dimensions, we will use the formula to calculate the midpoint of coordinates and :

We know that point has coordinates and we will let point have coordinates . The midpoint between these two points has coordinates .

Substituting these values into the formula, we have

We can then equate the individual components, giving us three equations to solve.

Firstly, the -coordinate gives us

Multiplying both sides of the equation by 2, we get

So,

Secondly, the -coordinate gives us

Multiplying both sides of the equation by 2, we get

So,

Finally, the -coordinate gives us

Multiplying both sides of the equation by 2, we get

So,

The coordinates of point are .

In two dimensions, we can calculate the distance between two points by using an adaption of the Pythagorean theorem. This states that , where is the length of the longest side, known as the hypotenuse, of a right triangle.

If two points and have coordinates and , respectively, then we can calculate the distance between them by using the following formula:

We will now consider how we can calculate the distance between two points in three dimensions.

Consider the three-dimensional rectangular prism , drawn below, and letβs assume that we want to travel from the bottom most left front corner, , to the topmost right back corner, .

First, letβs consider triangle on the bottom of the prism. The Pythagorean theorem tells us that .

So, .

Now, we make another triangle , with its base along and height .

We can use the Pythagorean theorem again such that . Substituting in the lengths of and , we see that .

Therefore, .

### Definition: The Distance between Two Points in 3D Space

If two points and have coordinates and , respectively, then we can calculate the distance between them by using the following formula:

This is an adaption of the Pythagorean theorem in three dimensions; we find the sum of the squares of the difference between each coordinate and then square root this answer.

In our final two questions, we will calculate the shortest distance between a point and one of the axes, as well as the distance between two points in space.

### Example 5: Finding the Distance between Two Points given Their Coordinates in Three Dimensions

Find the distance between the two points and .

In order to calculate the distance between two points in three dimensions, we will use the following formula, where the two points and have coordinates and respectively:

We will let point have coordinates and point have coordinates .

The distance between them is

The distance between the two points and is length units.

### Example 6: Finding the Distance between a Point and an Axis in 3D

What is the distance between the point and the -axis?

We know that any point will lie on the -axis if both its -coordinate and its -coordinate are equal to zero. This means we can define a point on the -axis as .

We recognize that the required distance is the perpendicular distance from the point to the -axis, which means the projection of the point on the -axis will be at the point .

The distance between two points can be calculated using the formula as follows

The distance between the point and the -axis is length units.

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

### Key Points

• Any point in three dimensions has coordinates written in the form .
• If the -coordinate is equal to zero, then we know that the point lies in the -plane; if the -coordinate is equal to zero, then we know that the point lies in the -plane; and if the -coordinate is equal to zero, then we know that the point lies in the -plane.
• If both the -coordinate and the -coordinate are equal to zero, then the point lies on the -axis; if both the -coordinate and the -coordinate are equal to zero, then the point lies on the -axis; and if both the -coordinate and the -coordinate are equal to zero, then the point lies on the -axis.
• The midpoint of two points with coordinates and lies at the point .
• We can also use the midpoint formula to calculate the endpoint of a line segment, given the midpoint and the other endpoint.
• The distance between two points with coordinates and is equal to .