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 π‘₯- and 𝑦-coordinate and can be written in the form (π‘₯,𝑦). 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 (0,0).

If two points 𝐴 and 𝐡 have coordinates (π‘₯,𝑦) and (π‘₯,𝑦), respectively, then we can calculate their midpoint by using the formula ο€Όπ‘₯+π‘₯2,𝑦+𝑦2.

If two points 𝐴 and 𝐡 have coordinates (π‘₯,𝑦) and (π‘₯,𝑦), respectively, then we can calculate the distance between them by using the distance formula, derived from the Pythagorean theorem, (π‘₯βˆ’π‘₯)+(π‘¦βˆ’π‘¦).

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 π‘₯-, 𝑦-, and 𝑧-coordinates and can be written in the form (π‘₯,𝑦,𝑧). 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 (βˆ’7,βˆ’8,0) lie?

  1. π‘₯𝑦
  2. π‘₯𝑧
  3. 𝑦𝑧

Answer

We know that a point in 3D will have π‘₯-, 𝑦-, and 𝑧-coordinates. In this question, π‘₯=βˆ’7, 𝑦=βˆ’8, and 𝑧=0.

As the 𝑧-coordinate is equal to zero, the point lies a distance of zero from the origin in the 𝑧-direction. This means that it will lie on the π‘₯𝑦-plane. In fact, any point with coordinates (π‘₯,𝑦,0) will lie on this plane.

We can therefore conclude that the point (βˆ’7,βˆ’8,0) lies on the π‘₯𝑦-plane.

Definition: The Three Coordinate Planes

Any point with coordinates (π‘₯,𝑦,0) will lie on the π‘₯𝑦-plane.

Similarly, any point with coordinates (π‘₯,0,𝑧) will lie on the π‘₯𝑧-plane, and any point with coordinates (0,𝑦,𝑧) will lie on the 𝑦𝑧-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 𝐴.

Answer

Any point in three dimensions will have π‘₯-, 𝑦-, and 𝑧-coordinates and can be written in the form (π‘₯,𝑦,𝑧).

Moving from the origin, we travel 3 units in the positive π‘₯-direction, βˆ’3 units in the 𝑦-direction, and finally 3 units in the 𝑧-direction.

This means that π‘₯=3, 𝑦=βˆ’3, and 𝑧=3.

The coordinates of point 𝐴 are (3,βˆ’3,3).

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 π‘₯-coordinates and the average of the 𝑦-coordinates. We will now extend this idea to three dimensions by finding the average of the 𝑧-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 𝐴 and 𝐡 have coordinates (π‘₯,𝑦,𝑧) and (π‘₯,𝑦,𝑧), respectively, then we can calculate their midpoint by using the following formula: ο€Όπ‘₯+π‘₯2,𝑦+𝑦2,𝑧+𝑧2.

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 𝐴 and 𝐡 have coordinates (8,βˆ’8,βˆ’12) and (βˆ’8,5,βˆ’8) respectively. Determine the coordinates of the midpoint of 𝐴𝐡.

Answer

In order to find the midpoint of two points in three dimensions, we will use the formula to calculate the midpoint of coordinates (π‘₯,𝑦,𝑧) and (π‘₯,𝑦,𝑧): ο€Όπ‘₯+π‘₯2,𝑦+𝑦2,𝑧+𝑧2.

We will let point 𝐴 have coordinates (π‘₯,𝑦,𝑧) and point 𝐡 have coordinates (π‘₯,𝑦,𝑧).

The midpoint between points 𝐴 and 𝐡 is =ο€½8+(βˆ’8)2,βˆ’8+52,βˆ’12+(βˆ’8)2=ο€Ό02,βˆ’32,βˆ’202=ο€Ό0,βˆ’32,βˆ’10.

The coordinates of the midpoint of 𝐴𝐡 are ο€Ό0,βˆ’32,βˆ’10.

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 (0,17,βˆ’10) is the midpoint of 𝐴𝐡 and that 𝐴(βˆ’19,7,14), what are the coordinates of 𝐡?

Answer

In order to find the midpoint of two points in three dimensions, we will use the formula to calculate the midpoint of coordinates (π‘₯,𝑦,𝑧) and (π‘₯,𝑦,𝑧) : ο€Όπ‘₯+π‘₯2,𝑦+𝑦2,𝑧+𝑧2.

We know that point 𝐴 has coordinates (βˆ’19,7,14) and we will let point 𝐡 have coordinates (π‘₯,𝑦,𝑧). The midpoint between these two points has coordinates (0,17,βˆ’10).

Substituting these values into the formula, we have (0,17,βˆ’10)=ο€Όβˆ’19+π‘₯2,7+𝑦2,14+𝑧2.

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

Firstly, the π‘₯-coordinate gives us 0=βˆ’19+π‘₯2.

Multiplying both sides of the equation by 2, we get 0=βˆ’19+π‘₯.

So, 19=π‘₯.

Secondly, the 𝑦-coordinate gives us 17=7+𝑦2.

Multiplying both sides of the equation by 2, we get 34=7+𝑦.

So, 27=𝑦.

Finally, the 𝑧-coordinate gives us βˆ’10=14+𝑧2.

Multiplying both sides of the equation by 2, we get βˆ’20=14+𝑧.

So, βˆ’34=𝑧.

The coordinates of point 𝐡 are (19,27,βˆ’34).

In two dimensions, we can calculate the distance between two points by using an adaption of the Pythagorean theorem. This states that π‘Ž+𝑏=π‘οŠ¨οŠ¨οŠ¨, where 𝑐 is the length of the longest side, known as the hypotenuse, of a right triangle.

If two points 𝐴 and 𝐡 have coordinates (π‘₯,𝑦) and (π‘₯,𝑦), respectively, then we can calculate the distance between them by using the following formula: (π‘₯βˆ’π‘₯)+(π‘¦βˆ’π‘¦).

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

Consider the three-dimensional rectangular prism 𝐴𝐡𝐢𝐷𝐸𝐹𝐺𝐻, drawn below, and let’s assume that we want to travel from the bottom most left front corner, 𝐴, to the topmost right back corner, 𝐺.

First, let’s consider triangle 𝐴𝐡𝐹 on the bottom of the prism. The Pythagorean theorem tells us that 𝐴𝐹=𝐴𝐡+𝐡𝐹.

So, 𝐴𝐹=√π‘₯+π‘¦οŠ¨οŠ¨.

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

We can use the Pythagorean theorem again such that 𝐴𝐺=𝐴𝐹+𝐹𝐺. Substituting in the lengths of 𝐴𝐹 and 𝐹𝐺, we see that 𝐴𝐺=ο„žο€»βˆšπ‘₯+𝑦+π‘§οŠ¨οŠ¨οŠ¨οŠ¨.

Therefore, 𝐴𝐺=√π‘₯+𝑦+π‘§οŠ¨οŠ¨οŠ¨.

Definition: The Distance between Two Points in 3D Space

If two points 𝐴 and 𝐡 have coordinates (π‘₯,𝑦,𝑧) and (π‘₯,𝑦,𝑧), respectively, then we can calculate the distance between them by using the following formula: (π‘₯βˆ’π‘₯)+(π‘¦βˆ’π‘¦)+(π‘§βˆ’π‘§).

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 𝐴(βˆ’7,12,3) and 𝐡(βˆ’4,βˆ’1,βˆ’8).

Answer

In order to calculate the distance between two points in three dimensions, we will use the following formula, where the two points 𝐴 and 𝐡 have coordinates (π‘₯,𝑦,𝑧) and (π‘₯,𝑦,𝑧) respectively: (π‘₯βˆ’π‘₯)+(π‘¦βˆ’π‘¦)+(π‘§βˆ’π‘§).

We will let point 𝐴 have coordinates (π‘₯,𝑦,𝑧) and point 𝐡 have coordinates (π‘₯,𝑦,𝑧).

The distance between them is =√(βˆ’4βˆ’(βˆ’7))+(βˆ’1βˆ’12)+(βˆ’8βˆ’3)=√(3)+(βˆ’13)+(βˆ’11)=√9+169+121=√299.

The distance between the two points 𝐴(βˆ’7,12,3) and 𝐡(βˆ’4,βˆ’1,βˆ’8) is √299 length units.

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

What is the distance between the point (19,5,5) and the π‘₯-axis?

Answer

We know that any point will lie on the π‘₯-axis if both its 𝑦-coordinate and its 𝑧-coordinate are equal to zero. This means we can define a point on the π‘₯-axis as (π‘₯,0,0).

We recognize that the required distance is the perpendicular distance from the point to the π‘₯-axis, which means the projection of the point on the π‘₯-axis will be at the point (19,0,0).

The distance between two points can be calculated using the formula (π‘₯βˆ’π‘₯)+(π‘¦βˆ’π‘¦)+(π‘§βˆ’π‘§) as follows √(19βˆ’19)+(5βˆ’0)+(5βˆ’0)=√0+(5)+(5)=√50=5√2.

The distance between the point (19,5,5) and the π‘₯-axis is 5√2 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 (π‘₯,𝑦,𝑧).
  • If the 𝑧-coordinate is equal to zero, then we know that the point lies in the π‘₯𝑦-plane; if the 𝑦-coordinate is equal to zero, then we know that the point lies in the π‘₯𝑧-plane; and if the π‘₯-coordinate is equal to zero, then we know that the point lies in the 𝑦𝑧-plane.
  • If both the 𝑦-coordinate and the 𝑧-coordinate are equal to zero, then the point lies on the π‘₯-axis; if both the π‘₯-coordinate and the 𝑧-coordinate are equal to zero, then the point lies on the 𝑦-axis; and if both the π‘₯-coordinate and the 𝑦-coordinate are equal to zero, then the point lies on the 𝑧-axis.
  • The midpoint of two points with coordinates (π‘₯,𝑦,𝑧) and (π‘₯,𝑦,𝑧) lies at the point ο€Όπ‘₯+π‘₯2,𝑦+𝑦2,𝑧+𝑧2.
  • 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 (π‘₯,𝑦,𝑧) and (π‘₯,𝑦,𝑧) is equal to (π‘₯βˆ’π‘₯)+(π‘¦βˆ’π‘¦)+(π‘§βˆ’π‘§).

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