Step by step
The question
Question: find the length of v = ⟨2, 2, 4⟩.
1 Draw it
Draw the three axes.
The arrow starts at the origin.
Go 2 along x, 2 along y, 4 along z.
The arrow goes from the origin to the end of that path.
2 Find the length
The length ‖v‖ is Pythagoras with three parts.
Square each part, add, then take the square root.
2 squared is 4.
2 squared is 4.
4 squared is 16.
Add: 4 plus 4 plus 16 is 24.
But 24 has a square number in it: 4.
24 is 4 times 6.
A root of a times splits into two roots.
√4 is 2.
So ‖v‖ = 2√6.
That is the exact answer, made simple.
With a calculator, that is about 4.9, to the nearest tenth.
Careful!
We don't add the parts.
The arrow is the long side of a right triangle, so we square, add, and take the root.
3 Check
Let's check it as Pythagoras twice.
Across the floor the square is 8.
Then up z: 8 plus 16 is 24.
The same!
Let's look back.
First: draw it.
Then: find the length.
Last: check.