Step by step
The question
Question: find the distance between (0, 4, 1) and (2, 0, 5).
1 Plot the points
Draw the three axes: x toward us, y right, z up.
Point A: (0, 4, 1).
Point B: (2, 0, 5).
The distance is the length of this straight segment.
2 The differences
Along x: x goes from 0 to 2.
2 minus 0 is 2.
Along y: y goes from 4 to 0.
0 minus 4 is −4.
The minus only says which way.
Up z: z goes from 1 to 5.
5 minus 1 is 4.
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.
2 squared is 4.
(−4) squared is 16.
4 squared is 16.
Minus times minus is plus.
4 Take the square root
Add: 4 plus 16 plus 16 is 36.
Careful!
36 is d squared.
We still take the square root.
6 times 6 is 36.
So d = 6.
5 Check
Let's check: square the distance.
We get 36, the same as 4 plus 16 plus 16.
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.