Step by step
The question
Question: find the distance between (3, 0, −1) and (5, 3, 5).
1 Plot the points
Draw the three axes: x toward us, y right, z up.
Point A: (3, 0, −1).
Point B: (5, 3, 5).
The distance is the length of this straight segment.
2 The differences
Along x: x goes from 3 to 5.
5 minus 3 is 2.
Along y: y goes from 0 to 3.
3 minus 0 is 3.
Up z: z goes from −1 to 5.
5 minus (−1) is 6.
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.
3 squared is 9.
6 squared is 36.
4 Take the square root
Add: 4 plus 9 plus 36 is 49.
Careful!
49 is d squared.
We still take the square root.
7 times 7 is 49.
So d = 7.
5 Check
Let's check: square the distance.
We get 49, the same as 4 plus 9 plus 36.
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.