### 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.