Step by step
The question
Question: find the magnitude of v = ⟨6, −3⟩.
1 Draw the arrow
Draw the grid.
The arrow starts at 0, at the origin.
It goes right 6 and down 3.
We call it v.
Across 6
and down 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.
6² is 6 times 6: 36.
(−3)² is 9.
Minus times minus is plus.
Write the squares: 36 + 9.
36 plus 9 is 45.
But 45 has a square number in it: 9.
45 is 9 times 5.
A root of a times splits into two roots.
√9 is 3.
So ‖v‖ = 3√5.
That is the exact answer, made simple.
Careful!
Not across plus up.
The arrow is the slanted side.
It is shorter than going across and then up.
3 Check
Let's check.
Square the length: we get 45.
The same as 36 plus 9.
Right!
Let's look back.
First: draw the arrow.
Then: find the length.
Last: check.