Step by step
The question
Question: find the magnitude of v = ⟨3, −4⟩.
v = ⟨3, −4⟩
1 Draw the arrow
Draw the grid.
The arrow starts at 0, at the origin.
It goes right 3 and down 4.
We call it v.
Across 3
and down 4.
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‖ = √32 + (−4)2
3² is 3 times 3: 9.
32 = 3 × 3 = 9
(−4)² is 16.
Minus times minus is plus.
(−4)2 = (−4) × (−4) = 16
Write the squares: 9 + 16.
‖v‖ = √9 + 16
9 plus 16 is 25.
‖v‖ = √25
5 times 5 is 25.
5 × 5 = 25
So ‖v‖ = 5.
‖v‖ = 5
Careful!
Not across plus up.
The arrow is the slanted side.
It is shorter than going across and then up.
Careful: not ‖v‖ = 3 + 4 = 7 ✗
3 Check
Let's check.
Square the length: we get 25.
The same as 9 plus 16.
Right!
5 × 5 = 25 and 9 + 16 = 25
Let's look back.
First: draw the arrow.
Then: find the length.
Last: check.
Answer
Careful: not ‖v‖ = 3 + 4 = 7 ✗