Step by step
The question
Question: find the unit vector in the direction of v = ⟨−12, −5⟩.
1 Draw the arrow
Draw the grid.
The arrow starts at the origin.
It goes left 12 and down 5.
Across 12 and down 5 make a right triangle.
2 Find the length
The length ‖v‖: the long side of the right triangle.
Pythagoras: square the two parts, add them, then take the square root.
Write the squares: 144 + 25.
144 plus 25 is 169.
13 times 13 is 169.
So ‖v‖ = 13.
3 Divide by the length
To make the length 1, divide each part by the length, 13.
The direction stays the same.
Careful!
Not 1 in each part.
The length must be 1, and the arrow must point the same way as v.
So u = ⟨−12/13, −5/13⟩.
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.
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.