Step by step
The question
Question: find the magnitude of v = ⟨2, 4⟩.
v = ⟨2, 4⟩
1 Draw the arrow
Draw the grid.
The arrow starts at 0, at the origin.
It goes right 2 and up 4.
We call it v.
Across 2
and up 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‖ = √22 + 42
2² is 2 times 2: 4.
22 = 2 × 2 = 4
4² is 16.
42 = 4 × 4 = 16
Write the squares: 4 + 16.
‖v‖ = √4 + 16
4 plus 16 is 20.
‖v‖ = √20
But 20 has a square number in it: 4.
20 is 4 times 5.
‖v‖ = √4 × 5
A root of a times splits into two roots.
‖v‖ = √4 × √5
√4 is 2.
2 × 2 = 4
So ‖v‖ = 2√5.
That is the exact answer, made simple.
‖v‖ = 2√5
Careful!
Not across plus up.
The arrow is the slanted side.
It is shorter than going across and then up.
Careful: not ‖v‖ = 2 + 4 = 6 ✗
3 Check
Let's check.
Square the length: we get 20.
The same as 4 plus 16.
Right!
(2√5)² = 4 × 5 = 20 and 4 + 16 = 20
Let's look back.
First: draw the arrow.
Then: find the length.
Last: check.
Answer
Careful: not ‖v‖ = 2 + 4 = 6 ✗