Step by step
The question
Question: find the length of (2, 4, 2).
v = [2; 4; 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‖ = √22 + 42 + 22
Careful!
The entries are squared first, then added.
Adding the entries themselves is not the length.
Careful: not 2 + 4 + 2 = 8 ✗
The squares: 4 + 16 + 4.
‖v‖ = √4 + 16 + 4
Added: 24.
‖v‖ = √24
2 Take the square root
24 has a square inside: 4 × 6.
The root of 4 is 2.
24 = 4 × 6
So the length is 2√6, exactly.
‖v‖ = 2√6
3 Check
Let's check: the length squared must give back 24.
It does.
So the answer is right!
(2√6)² = 2² × 6 = 4 × 6 = 24
Let's look back.
First: square and add.
Then: take the square root.
Last: check.
Answer
‖v‖ = 2√6