Step by step
The question
Question: find the magnitude of v = ⟨−12, 5⟩.
v = ⟨−12, 5⟩
1 Draw the arrow
Draw the grid.
The arrow starts at 0, at the origin.
It goes left 12 and up 5.
We call it v.
Across 12
and up 5.
They make a right angle.
The arrow is the longest side.
2 Find the length
Pythagoras: square the two parts, add them, then take the square root.
‖v‖ = √(−12)2 + 52
(−12)² is (−12) times (−12): 144.
Minus times minus is plus.
(−12)2 = (−12) × (−12) = 144
5² is 25.
52 = 5 × 5 = 25
Write the squares: 144 + 25.
‖v‖ = √144 + 25
144 plus 25 is 169.
‖v‖ = √169
13 times 13 is 169.
13 × 13 = 169
So ‖v‖ = 13.
‖v‖ = 13
Careful!
Not across plus up.
The arrow is the slanted side.
It is shorter than going across and then up.
Careful: not ‖v‖ = 12 + 5 = 17 ✗
3 Check
Let's check.
Square the length: we get 169.
The same as 144 plus 25.
Right!
13 × 13 = 169 and 144 + 25 = 169
Let's look back.
First: draw the arrow.
Then: find the length.
Last: check.
Answer
Careful: not ‖v‖ = 12 + 5 = 17 ✗