Step by step
The question
Question: find the distance between (−1, 2, 0) and (3, 2, 3).
1 Plot the points
Draw the three axes: x toward us, y right, z up.
Point A: (−1, 2, 0).
Point B: (3, 2, 3).
The distance is the length of this straight segment.
2 The differences
Along x: x goes from −1 to 3.
3 minus (−1) is 4.
Along y: y goes from 2 to 2.
2 minus 2 is 0.
No move that way.
Up z: z goes from 0 to 3.
3 minus 0 is 3.
3 Square and add
On the flat, the move 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.
4 squared is 16.
0 squared is 0.
3 squared is 9.
4 Take the square root
Add: 16 plus 0 plus 9 is 25.
Careful!
25 is d squared.
We still take the square root.
5 times 5 is 25.
So d = 5.
5 Check
Let's check: square the distance.
We get 25, the same as 16 plus 0 plus 9.
Right!
Going from B to A instead only changes the signs.
The squares are the same.
Right!
Let's look back.
First: plot the points.
Then: the differences.
Then: square and add.
Then: take the square root.
Last: check.