Step by step
The question
Question: find the magnitude of v = ⟨−1, 3⟩.
v = ⟨−1, 3⟩
1 Draw the arrow
Draw the grid.
The arrow starts at 0, at the origin.
It goes left 1 and up 3.
We call it v.
Across 1
and up 3.
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‖ = √(−1)2 + 32
(−1)² is (−1) times (−1): 1.
Minus times minus is plus.
(−1)2 = (−1) × (−1) = 1
3² is 9.
32 = 3 × 3 = 9
Write the squares: 1 + 9.
‖v‖ = √1 + 9
1 plus 9 is 10.
‖v‖ = √10
10 is not a square number.
So we keep √10.
That is the exact answer.
‖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‖ = 1 + 3 = 4 ✗
3 Check
Let's check.
Square the length: we get 10.
The same as 1 plus 9.
Right!
√10 × √10 = 10 and 1 + 9 = 10
Let's look back.
First: draw the arrow.
Then: find the length.
Last: check.
Answer
Careful: not ‖v‖ = 1 + 3 = 4 ✗