Step by step
The question
Question: find u − v, when u = ⟨3, −1, 2⟩ and v = ⟨1, 4, −2⟩.
u = ⟨3, −1, 2⟩, v = ⟨1, 4, −2⟩
1 Take away part by part
Take away the matching parts: x with x, y with y, z with z.
u − v = ⟨3 − 1, −1 − 4, 2 − (−2)⟩
Careful!
Taking away a minus number is adding.
2 minus (−2) is 4.
Careful: not 2 − (−2) = 0 ✗
So u − v = ⟨2, −5, 4⟩.
u − v = ⟨2, −5, 4⟩
2 Check
Let's check with a picture.
First u from the origin.
Then −v from its tip.
The arrow from the start to the end is the answer.
Tip to tail: ⟨3, −1, 2⟩ then ⟨−1, −4, 2⟩ ends at ⟨2, −5, 4⟩.
Right!
⟨3, −1, 2⟩ + ⟨−1, −4, 2⟩ = ⟨2, −5, 4⟩
Let's look back.
First: take away part by part.
Last: check.
Answer
u − v = ⟨2, −5, 4⟩