Step by step
The question
Question: find the unit vector in the direction of v = ⟨0, −5⟩.
v = ⟨0, −5⟩
1 Draw the arrow
Draw the grid.
The arrow starts at the origin.
It goes down 5.
2 Find the length
The arrow goes straight down.
So its length is just 5.
‖v‖ = 5
3 Divide by the length
To make the length 1, divide each part by the length, 5.
The direction stays the same.
u = ⟨05, −55⟩
Careful!
Not 1 in each part.
The length must be 1, and the arrow must point the same way as v.
Careful: not ⟨1, 1⟩ ✗
Make the fractions simpler.
So u = ⟨0, −1⟩.
u = ⟨0, −1⟩
On the grid, u is a short arrow along v: length 1.
4 Check
Let's check: the length of u must be 1.
Square the parts and add them.
(−1)2 = 1 = 1
We get 1.
So the length of u is 1.
Right!
Let's look back.
First: draw the arrow.
Then: find the length.
Then: divide by the length.
Last: check.
Answer
u = ⟨0, −1⟩