Lesson Video: Points, Midpoints, and Distances in Space Mathematics

In this video, 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.

16:13

Video Transcript

In this video, we will learn how to find the coordinates of a point in three dimensions. We will also calculate the distance between two points in 3D and then midpoint. We will begin by recalling what we know about points, midpoints, and distances in two dimensions. The two-dimensional 𝑥𝑦-coordinate plane is drawn below. Any point on this coordinate plane will have an 𝑥- and 𝑦-coordinates. Let’s consider the two points 𝐴 and 𝐵 with coordinates 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two, respectively.

In order to find the midpoint of 𝐴 and 𝐵, we find the average of the 𝑥- and 𝑦-coordinates. The 𝑥-coordinate of the midpoint will be equal to 𝑥 one plus 𝑥 two divided by two. And the 𝑦-coordinate will be equal to 𝑦 one plus 𝑦 two over two. In order to calculate the distance between two points on the 𝑥𝑦-plane, we use an adaption of the Pythagorean theorem. The distance between point 𝐴 and point 𝐵 is the square root of 𝑥 two minus 𝑥 one squared plus 𝑦 two minus 𝑦 one squared. We find the difference between the 𝑥-coordinates and square the answer. We then find the difference between the 𝑦-coordinates and square this answer. The sum of these square rooted is the distance between the two points on the 𝑥𝑦-plane.

We will now look at how we can adapt these two formulas when dealing in three dimensions. The three-dimensional 𝑥𝑦𝑧-plane could be drawn in many ways on a two-dimensional surface. We know that any point will have an 𝑥-, 𝑦-, and 𝑧-coordinates. For example, the two points shown have coordinates 𝑥 one, 𝑦 one, 𝑧 one and 𝑥 two, 𝑦 two, 𝑧 two. We can find the midpoint of 𝐴 and 𝐵 by finding the average of the 𝑥-, 𝑦-, and 𝑧-coordinates. The 𝑥-coordinate of the midpoint will be equal to 𝑥 one plus 𝑥 two divided by two. The 𝑦-coordinate will be 𝑦 one plus 𝑦 two divided by two. And the 𝑧-coordinate will be 𝑧 one plus 𝑧 two divided by two.

We can extend the distance formula in the same way. The distance between two points in three dimensions is equal to the square root of 𝑥 two minus 𝑥 one squared plus 𝑦 two minus 𝑦 one squared plus 𝑧 two minus 𝑧 one squared. We simply repeat the process used with the 𝑥- and 𝑦-coordinates with the 𝑧-coordinate. We will now look at some questions where we need to identify points in three dimensions.

In which of the following coordinate planes does the point negative seven, negative eight, zero lie? Is it (A) the 𝑥𝑦-plane, (B) the 𝑥𝑧-plane, or (C) the 𝑦𝑧-plane?

We know that any point in three dimensions has an 𝑥-, 𝑦-, and 𝑧-coordinate. In this question, the 𝑥-coordinate is negative seven, the 𝑦-coordinate is negative eight, and the 𝑧-coordinate is zero. As 𝑧 is equal to zero, the point will not move in the direction of the 𝑧-axis. We can therefore conclude that as 𝑧 is equal to zero, the point will lie on the 𝑥𝑦-plane. If our 𝑦-coordinate was equal to zero but 𝑥 and 𝑧 had a positive or negative value, the point would lie in the 𝑥𝑧-plane. In a similar way, a point would lie in the 𝑦𝑧-plane if it had coordinate zero, 𝑦, 𝑧, where 𝑦 and 𝑧 are positive or negative values.

In our next question, we need to find the coordinates of a point graphically.

Determine the coordinates of point 𝐴.

Any point on the 3D plane will have an 𝑥-, 𝑦-, and 𝑧-coordinate. We can see from our diagram that point 𝐴 has an 𝑥-coordinate of three. It has a 𝑦-coordinate of negative three. Finally, it has a 𝑧-coordinate of three. We can therefore conclude that the coordinates of point 𝐴 are three, negative three, three. If we weren’t able to spot this immediately on our diagram, we could begin by considering the point 𝐵 in the two-dimensional 𝑥𝑦-plane. Point 𝐵 has an 𝑥-coordinate equal to three and a 𝑦-coordinate equal to negative three. As it lies on the 𝑥𝑦-plane, it will have a 𝑧-coordinate equal to zero.

Point 𝐴 lies directly above point 𝐵. This means its 𝑥- and 𝑦-coordinates will be the same. All we now need to work out is the distance traveled along the 𝑧-axis to get from point 𝐵 to point 𝐴. As this is equal to three, the 𝑧-coordinate of point 𝐴 is three. This confirms that point 𝐴 has coordinates three, negative three, three.

In our next question, we need to work out the midpoint of a line segment.

Points 𝐴 and 𝐵 have coordinates eight, negative eight, negative 12 and negative eight, five, negative eight, respectively. Determine the coordinates of the midpoint of line segment 𝐴𝐵.

We recall that in order to find the midpoint of two points in three dimensions, we find the average of the 𝑥-, 𝑦-, and 𝑧-coordinates. We can begin by letting point 𝐴 have coordinates 𝑥 one, 𝑦 one, 𝑧 one and point 𝐵: 𝑥 two, 𝑦 two, 𝑧 two. The 𝑥-coordinate of our midpoint will be equal to eight plus negative eight divided by two. Eight plus negative eight is equal to zero and zero divided by two is equal to zero. The 𝑦-coordinates of 𝐴 and 𝐵 are negative eight and five. This means that the 𝑦-coordinate of the midpoint will be equal to negative eight plus five divided by two. This is equal to negative three over two, which we could write as negative one and a half or negative 1.5. We will leave the answer as a top heavy or improper fraction.

The 𝑧-coordinate of our midpoint is equal to negative 12 plus negative eight divided by two. Negative 12 plus negative eight is equal to negative 20. Dividing this by two gives us negative 10. The midpoint of the line segment 𝐴𝐵 has coordinates zero, negative three over two, negative 10. We could check this answer by looking at the distances between these values and the corresponding values in points 𝐴 and 𝐵. Zero is eight away from both eight and negative eight. Negative three over two or negative 1.5 is 6.5 away from negative eight and also from five. Finally, negative 10 is two away from negative 12 and also two away from negative eight. This confirms that the midpoint of points 𝐴 and 𝐵 is zero, negative three over two, negative 10.

In our next question, we’ll need to find the distance between a point and one of the axes.

What is the distance between the point 19, five, five and the 𝑥-axis?

Any point that lies on the 𝑥-axis will have coordinates 𝑥, zero, zero. Both the 𝑦- and 𝑧-coordinates must be equal to zero. We’re given the coordinates of a point 19, five, five. The point on the 𝑥-axis that is closest to this will have coordinates 19, zero, zero. The shortest distance will be to the point where the 𝑥-coordinate is the same. We know that we can calculate the distance between two points in three dimensions using an adaption of the Pythagorean theorem. If we have two points with coordinates 𝑥 one, 𝑦 one, 𝑧 one and 𝑥 two, 𝑦 two, 𝑧 two, the distance between them is equal to the square root of 𝑥 two minus 𝑥 one squared plus 𝑦 two minus 𝑦 one squared plus 𝑧 two minus 𝑧 one squared.

Substituting in our two coordinates gives us the square root of 19 minus 19 squared plus zero minus five squared plus zero minus five squared. 19 minus 19 is equal to zero. Zero minus five is equal to negative five. So we are left with the square root of negative five squared plus negative five squared. Multiplying a negative number by a negative number gives us a positive answer. Therefore, negative five squared is equal to 25. This means that our answer simplifies to the square root of 50.

It is worth pointing out that we could have subtracted the coordinates in the other order as five minus zero squared is also equal to 25. As squaring a number always gives a positive answer, it doesn’t matter which order we subtract our coordinates in. We can actually simplify our answer by using our laws of radicals or surds. The square root of 50 is equal to the square root of 25 multiplied by the square root of two. As the square root of 25 equals five, we’re left with five multiplied by the square root of two or five root two. The square root of 50 is equal to five root two. We can therefore conclude that the distance between the points 19, five, five and the 𝑥-axis is five root two length units.

We might actually notice a shortcut here. To find the distance between any point and an axis, we simply find the sum of the squares of the other two coordinates and then square root the answer. As we want to calculate the distance to the 𝑥-axis, we square the 𝑦- and 𝑧-coordinates, find their sum, and then square root our answer. If we needed to calculate the distance between a point and the 𝑦-axis, we would square the 𝑥- and 𝑧-coordinates, find the sum of these, and then square root that answer. We would use the same method to find the distance between a point and the 𝑧-axis, this time using the 𝑥- and 𝑦-coordinates.

In our final question, we will find the distance between two points given their coordinates in three dimensions.

Find the distance between the two points 𝐴: negative seven, 12, three and 𝐵: negative four, negative one, negative eight.

We know that we can find the distance between two points in three-dimensional space using the following formula. The distance is equal to the square root of 𝑥 two minus 𝑥 one squared plus 𝑦 two minus 𝑦 one squared plus 𝑧 two minus 𝑧 one squared. In this question, we will let the coordinates of point 𝐴 be 𝑥 one, 𝑦 one, 𝑧 one and the coordinates of point 𝐵 be 𝑥 two, 𝑦 two, 𝑧 two. Substituting in these values gives us the square root of negative four minus negative seven squared plus negative one minus 12 squared plus negative eight minus three squared.

Negative four minus negative seven is the same as negative four plus seven. This is equal to three. Negative one minus 12 is equal to negative 13. Finally, negative eight minus three is equal to negative 11. We know that squaring a negative number gives a positive answer. This means that three squared is equal to nine, negative 13 squared is 169, and negative 11 squared is 121. 169 plus 121 is equal to 290. And adding nine to this gives us 299. We can therefore conclude that the distance between the two points negative seven, 12, three and negative four, negative one, negative eight is the square root of 299 length units.

We will now summarize the key points from this video. In this video, we saw that any point in three dimensions has coordinates 𝑥, 𝑦, and 𝑧. We saw that if our 𝑧-coordinate is equal to zero, the point lies on the 𝑥𝑦-plane. If the 𝑦-coordinate was equal to zero, it would lie on the 𝑥𝑧-plane. In the same way, if 𝑥 was equal to zero, the point would lie on the 𝑦𝑧-plane. We also saw that if a point has two coordinates that are equal to zero, for example, if 𝑦 equals zero and 𝑧 equals zero, it will lie on one of the axes, in this case, the 𝑥-axis.

If 𝑥 and 𝑦 were both equal to zero, the point would lie on the 𝑧-axis. And in the same way, if 𝑥 and 𝑧 were equal to zero, the point would lie on the 𝑦-axis. We saw that the midpoint of two points 𝐴 and 𝐵 has coordinates 𝑥 one plus 𝑥 two over two, 𝑦 one plus 𝑦 two over two, and 𝑧 one plus 𝑧 two over two. We find the average of the 𝑥-, 𝑦-, and 𝑧-coordinates.

We also saw that we can calculate the distance between the same two points by square rooting 𝑥 two minus 𝑥 one squared plus 𝑦 two minus 𝑦 one squared plus 𝑧 two minus 𝑧 one squared. These two formulas will allow us to solve practical problems involving coordinates in three dimensions.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.