Step by step
The question
Question: find the distance between (2, −1, 3) and (4, 3, 7).
1 Plot the points
Draw the three axes: x toward us, y right, z up.
Point A: (2, −1, 3).
Point B: (4, 3, 7).
The distance is the length of this straight segment.
2 The differences
Along x: x goes from 2 to 4.
4 minus 2 is 2.
Along y: y goes from −1 to 3.
3 minus (−1) is 4.
Up z: z goes from 3 to 7.
7 minus 3 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.
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.