Step by step
The question
Question: find the magnitude of v = ⟨6, 8⟩.
v = ⟨6, 8⟩
1 Draw the arrow
Draw the grid.
The arrow starts at 0, at the origin.
It goes right 6 and up 8.
We call it v.
Across 6
and up 8.
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‖ = √62 + 82
6² is 6 times 6: 36.
62 = 6 × 6 = 36
8² is 64.
82 = 8 × 8 = 64
Write the squares: 36 + 64.
‖v‖ = √36 + 64
36 plus 64 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‖ = 6 + 8 = 14 ✗
3 Check
Let's check.
Square the length: we get 100.
The same as 36 plus 64.
Right!
10 × 10 = 100 and 36 + 64 = 100
Let's look back.
First: draw the arrow.
Then: find the length.
Last: check.
Answer
Careful: not ‖v‖ = 6 + 8 = 14 ✗