Step by step
The question
Question: find the unit vector in the direction of v = ⟨6, 2, 3⟩.
v = ⟨6, 2, 3⟩
1 Find the length
The length ‖v‖ is Pythagoras with three parts.
Square each part, add, then take the square root.
‖v‖ = √62 + 22 + 32
6 squared is 36.
2 squared is 4.
3 squared is 9.
‖v‖ = √36 + 4 + 9
Add: 36 plus 4 plus 9 is 49.
‖v‖ = √49
7 times 7 is 49.
7 × 7 = 49
So ‖v‖ = 7.
‖v‖ = 7
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 7.
u = ⟨67, 27, 37⟩
So the unit vector is u = ⟨6/7, 2/7, 3/7⟩.
u = ⟨67, 27, 37⟩
Careful!
We divide by the length.
We don't take it away.
Careful: not ⟨6, 2, 3⟩ − 7 ✗
3 Check
Let's check: the length of u.
Square each part: over 49.
They add up to 49/49, which is 1.
Right!
‖u‖ = √(36/49 + 4/49 + 9/49) = √(49/49) = 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 ⟨6, 2, 3⟩ − 7 ✗