Step by step
The question
Question: find the unit vector in the direction of v = ⟨2, −1, 2⟩.
v = ⟨2, −1, 2⟩
1 Find the length
The length ‖v‖ is Pythagoras with three parts.
Square each part, add, then take the square root.
‖v‖ = √22 + (−1)2 + 22
2 squared is 4.
(−1) squared is 1.
2 squared is 4.
Minus times minus is plus.
‖v‖ = √4 + 1 + 4
Add: 4 plus 1 plus 4 is 9.
‖v‖ = √9
3 times 3 is 9.
3 × 3 = 9
So ‖v‖ = 3.
‖v‖ = 3
2 Divide by the length
To make the length 1, we divide the vector by its own length.
Then it keeps its direction.
u = v‖v‖
length ÷ length = 1
Divide each part by 3.
u = ⟨23, −13, 23⟩
So the unit vector is u = ⟨2/3, −1/3, 2/3⟩.
u = ⟨23, −13, 23⟩
Careful!
We divide by the length.
We don't take it away.
Careful: not ⟨2, −1, 2⟩ − 3 ✗
3 Check
Let's check: the length of u.
Square each part: over 9.
They add up to 9/9, which is 1.
Right!
‖u‖ = √(4/9 + 1/9 + 4/9) = √(9/9) = 1
And every part has the same sign as in v, so it points the same way.
Right!
same signs as v → same direction
Let's look back.
First: find the length.
Then: divide by the length.
Last: check.
Answer
Careful: not ⟨2, −1, 2⟩ − 3 ✗