Step by step
The question
Question: find the magnitude of v = ⟨−8, 6⟩.
v = ⟨−8, 6⟩
1 Draw the arrow
Draw the grid.
The arrow starts at 0, at the origin.
It goes left 8 and up 6.
We call it v.
Across 8
and up 6.
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‖ = √(−8)2 + 62
(−8)² is (−8) times (−8): 64.
Minus times minus is plus.
(−8)2 = (−8) × (−8) = 64
6² is 36.
62 = 6 × 6 = 36
Write the squares: 64 + 36.
‖v‖ = √64 + 36
64 plus 36 is 100.
‖v‖ = √100
10 times 10 is 100.
10 × 10 = 100
So ‖v‖ = 10.
‖v‖ = 10
Careful!
Not across plus up.
The arrow is the slanted side.
It is shorter than going across and then up.
Careful: not ‖v‖ = 8 + 6 = 14 ✗
3 Check
Let's check.
Square the length: we get 100.
The same as 64 plus 36.
Right!
10 × 10 = 100 and 64 + 36 = 100
Let's look back.
First: draw the arrow.
Then: find the length.
Last: check.
Answer
Careful: not ‖v‖ = 8 + 6 = 14 ✗