Step by step
The question
Question: find the distance between (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.
Along y: y goes from 2 to 6.
6 minus 2 is 4.
Up z: z goes from 3 to 15.
15 minus 3 is 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.
3 squared is 9.
4 squared is 16.
12 squared is 144.
4 Take the square root
Add: 9 plus 16 plus 144 is 169.
Careful!
169 is d squared.
We still take the square root.
13 times 13 is 169.
So d = 13.
5 Check
Let's check: square the distance.
We get 169, the same as 9 plus 16 plus 144.
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.