Step by step
The question
Question: find the length of (1, 2, 2).
v = [1; 2; 2]
The length of v, written with double bars, is Pythagoras in more directions: square every entry, add, then take the square root.
‖v‖ = √(entries squared, added)
1 Square and add
Every entry squared.
‖v‖ = √12 + 22 + 22
Careful!
The entries are squared first, then added.
Adding the entries themselves is not the length.
Careful: not 1 + 2 + 2 = 5 ✗
The squares: 1 + 4 + 4.
‖v‖ = √1 + 4 + 4
Added: 9.
‖v‖ = √9
2 Take the square root
9 is 3 squared.
So the length is 3.
‖v‖ = 3
3 Check
Let's check: the length squared must give back 9.
It does.
So the answer is right!
(3)² = 9
Let's look back.
First: square and add.
Then: take the square root.
Last: check.
Answer
‖v‖ = 3