Step by step
The question
Question: find the distance between (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.
Along y: y goes from 2 to 4.
4 minus 2 is 2.
Up z: z goes from 3 to 6.
6 minus 3 is 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.
1 squared is 1.
2 squared is 4.
3 squared is 9.
4 Take the square root
Add: 1 plus 4 plus 9 is 14.
Careful!
14 is d squared.
We still take the square root.
14 is not a square number, so √14 is the exact answer.
With a calculator, that is about 3.7, to the nearest tenth.
5 Check
Let's check: square the distance.
We get 14, the same as 1 plus 4 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.