Step by step
The question
Question: find the unit vector in the direction of v = ⟨−4, 4, 2⟩.
1 Find the length
The length ‖v‖ is Pythagoras with three parts.
Square each part, add, then take the square root.
(−4) squared is 16.
4 squared is 16.
2 squared is 4.
Minus times minus is plus.
Add: 16 plus 16 plus 4 is 36.
6 times 6 is 36.
So ‖v‖ = 6.
2 Divide by the length
To make the length 1, we divide the vector by its own length.
Then it keeps its direction.
Divide each part by 6.
Make the fractions simpler where we can.
So the unit vector is u = ⟨−2/3, 2/3, 1/3⟩.
Careful!
We divide by the length.
We don't take it away.
3 Check
Let's check: the length of u.
Square each part: over 36.
They add up to 36/36, which is 1.
Right!
And every part has the same sign as in v, so it points the same way.
Right!
Let's look back.
First: find the length.
Then: divide by the length.
Last: check.