Step by step
The question
Question: find 2u − 3v, when u = ⟨1, −2, 4⟩ and v = ⟨2, 1, −3⟩.
u = ⟨1, −2, 4⟩, v = ⟨2, 1, −3⟩
1 Multiply by the numbers
2u means 2 times every part of u.
2u = ⟨2 × 1, 2 × (−2), 2 × 4⟩
2 times 1 is 2.
2 times (−2) is −4.
2 times 4 is 8.
2u = ⟨2, −4, 8⟩
3v means 3 times every part of v.
3v = ⟨3 × 2, 3 × 1, 3 × (−3)⟩
3 times 2 is 6.
3 times 1 is 3.
3 times (−3) is −9.
3v = ⟨6, 3, −9⟩
2 Take away part by part
Take away the matching parts: x with x, y with y, z with z.
2u − 3v = ⟨2 − 6, −4 − 3, 8 − (−9)⟩
Careful!
Taking away a minus number is adding.
8 minus (−9) is 17.
Careful: not 8 − (−9) = −1 ✗
So 2u − 3v = ⟨−4, −7, 17⟩.
2u − 3v = ⟨−4, −7, 17⟩
3 Check
Let's check with a picture.
First 2u from the origin.
Then −3v from its tip.
The arrow from the start to the end is the answer.
Tip to tail: ⟨2, −4, 8⟩ then ⟨−6, −3, 9⟩ ends at ⟨−4, −7, 17⟩.
Right!
⟨2, −4, 8⟩ + ⟨−6, −3, 9⟩ = ⟨−4, −7, 17⟩
Let's look back.
First: multiply by the numbers.
Then: take away part by part.
Last: check.
Answer
2u − 3v = ⟨−4, −7, 17⟩