Step by step
The question
Question: find the length of (1, −2, 3, 1).
v = [1; −2; 3; 1]
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, a negative one in brackets.
‖v‖ = √12 + (−2)2 + 32 + 12
Careful!
The entries are squared first, then added.
Adding the entries themselves is not the length.
Careful: not 1 + −2 + 3 + 1 = 3 ✗
The squares: 1 + 4 + 9 + 1.
‖v‖ = √1 + 4 + 9 + 1
Added: 15.
‖v‖ = √15
2 Take the square root
3 Check
Let's check: the length squared must give back 15.
It does.
So the answer is right!
(√15)² = 15
Let's look back.
First: square and add.
Then: take the square root.
Last: check.
Answer
‖v‖ = √15