Step by step
The question
Question: find the unit vector in the direction of v = ⟨3, 4, 12⟩.
v = ⟨3, 4, 12⟩
1 Find the length
The length ‖v‖ is Pythagoras with three parts.
Square each part, add, then take the square root.
‖v‖ = √32 + 42 + 122
3 squared is 9.
4 squared is 16.
12 squared is 144.
‖v‖ = √9 + 16 + 144
Add: 9 plus 16 plus 144 is 169.
‖v‖ = √169
13 times 13 is 169.
13 × 13 = 169
So ‖v‖ = 13.
‖v‖ = 13
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 13.
u = ⟨313, 413, 1213⟩
So the unit vector is u = ⟨3/13, 4/13, 12/13⟩.
u = ⟨313, 413, 1213⟩
Careful!
We divide by the length.
We don't take it away.
Careful: not ⟨3, 4, 12⟩ − 13 ✗
3 Check
Let's check: the length of u.
Square each part: over 169.
They add up to 169/169, which is 1.
Right!
‖u‖ = √(9/169 + 16/169 + 144/169) = √(169/169) = 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 ⟨3, 4, 12⟩ − 13 ✗