Worked examples
The question
Question: plot the point (2, 3, 4).
(2, 3, 4)
The numbers always come in this order: x, then y, then z.
= (x, y, z)
So x = 2, y = 3 and z = 4.
x = 2 y = 3 z = 4
1 Set up the grid
Three axes, at square corners to each other.
x comes toward us, y goes right, z goes up.
The floor is flat, where z is 0.
z is the height above it.
2 Plot the point
Start at the origin, where the axes meet.
Go 2 along x.
Then go 3 along y.
Then go up 4, straight up, along z.
That is the point (2, 3, 4).
Careful!
The order matters.
The first number is always x.
(3, 2, 4) is a different point.
Careful: x first, then y, then z
3 Check
Let's check.
Drop straight down to the floor.
On the floor we are at x = 2, y = 3.
The height is 4.
It matches!
floor: (2, 3) height: 4
Let's look back.
First: set up the grid.
Then: plot the point.
Last: check.
▶ Watch it solved
The question
Question: find the distance between (1, 2, 3) and (4, 6, 15).
(1, 2, 3) and (4, 6, 15)
1 Plot the points
Draw the three axes: x toward us, y right, z up.
Point A: (1, 2, 3).
Point B: (4, 6, 15).
The distance is the length of this straight segment.
2 The differences
Along x: x goes from 1 to 4.
4 minus 1 is 3.
4 − 1 = 3
Along y: y goes from 2 to 6.
6 minus 2 is 4.
6 − 2 = 4
Up z: z goes from 3 to 15.
15 minus 3 is 12.
15 − 3 = 12
3 Square and add
On the flat, the x and y moves make a right triangle.
Its long side is this green line.
The green line and the move up z make a second right triangle.
Its long side is AB.
So Pythagoras works twice.
Square the three differences, add them, then take the square root.
d = √32 + 42 + 122
3 squared is 9.
4 squared is 16.
12 squared is 144.
d = √9 + 16 + 144
4 Take the square root
Add: 9 plus 16 plus 144 is 169.
d = √169
Careful!
169 is d squared.
We still take the square root.
Careful: not d = 169 ✗
13 times 13 is 169.
13 × 13 = 169
So d = 13.
d = 13
5 Check
Let's check: square the distance.
We get 169, the same as 9 plus 16 plus 144.
Right!
13² = 169 and 9 + 16 + 144 = 169
Going from B to A instead only changes the signs.
The squares are the same.
Right!
(1 − 4)² = 9
Let's look back.
First: plot the points.
Then: the differences.
Then: square and add.
Then: take the square root.
Last: check.
▶ Watch it solved
The question
Question: find the distance between (1, 2, 3) and (2, 4, 6).
(1, 2, 3) and (2, 4, 6)
1 Plot the points
Draw the three axes: x toward us, y right, z up.
Point A: (1, 2, 3).
Point B: (2, 4, 6).
The distance is the length of this straight segment.
2 The differences
Along x: x goes from 1 to 2.
2 minus 1 is 1.
2 − 1 = 1
Along y: y goes from 2 to 4.
4 minus 2 is 2.
4 − 2 = 2
Up z: z goes from 3 to 6.
6 minus 3 is 3.
6 − 3 = 3
3 Square and add
On the flat, the x and y moves make a right triangle.
Its long side is this green line.
The green line and the move up z make a second right triangle.
Its long side is AB.
So Pythagoras works twice.
Square the three differences, add them, then take the square root.
d = √12 + 22 + 32
1 squared is 1.
2 squared is 4.
3 squared is 9.
d = √1 + 4 + 9
4 Take the square root
Add: 1 plus 4 plus 9 is 14.
d = √14
Careful!
14 is d squared.
We still take the square root.
Careful: not d = 14 ✗
14 is not a square number, so √14 is the exact answer.
d = √14
With a calculator, that is about 3.7, to the nearest tenth.
d ≈ 3.7
5 Check
Let's check: square the distance.
We get 14, the same as 1 plus 4 plus 9.
Right!
d² = 14 and 1 + 4 + 9 = 14
Going from B to A instead only changes the signs.
The squares are the same.
Right!
(1 − 2)² = 1
Let's look back.
First: plot the points.
Then: the differences.
Then: square and add.
Then: take the square root.
Last: check.
▶ Watch it solved
