Step by step
The question
Question: find 3u − 2v, when u = ⟨2, 1, 0⟩ and v = ⟨−1, 3, 2⟩.
u = ⟨2, 1, 0⟩, v = ⟨−1, 3, 2⟩
1 Multiply by the numbers
3u means 3 times every part of u.
3u = ⟨3 × 2, 3 × 1, 3 × 0⟩
3 times 2 is 6.
3 times 1 is 3.
3 times 0 is 0.
3u = ⟨6, 3, 0⟩
2v means 2 times every part of v.
2v = ⟨2 × (−1), 2 × 3, 2 × 2⟩
2 times (−1) is −2.
2 times 3 is 6.
2 times 2 is 4.
2v = ⟨−2, 6, 4⟩
2 Take away part by part
Take away the matching parts: x with x, y with y, z with z.
3u − 2v = ⟨6 − (−2), 3 − 6, 0 − 4⟩
Careful!
Taking away a minus number is adding.
6 minus (−2) is 8.
Careful: not 6 − (−2) = 4 ✗
So 3u − 2v = ⟨8, −3, −4⟩.
3u − 2v = ⟨8, −3, −4⟩
3 Check
Let's check with a picture.
First 3u from the origin.
Then −2v from its tip.
The arrow from the start to the end is the answer.
Tip to tail: ⟨6, 3, 0⟩ then ⟨2, −6, −4⟩ ends at ⟨8, −3, −4⟩.
Right!
⟨6, 3, 0⟩ + ⟨2, −6, −4⟩ = ⟨8, −3, −4⟩
Let's look back.
First: multiply by the numbers.
Then: take away part by part.
Last: check.
Answer
3u − 2v = ⟨8, −3, −4⟩