Step by step
The question
Question: find 2u + v, when u = ⟨1, 2, 3⟩ and v = ⟨4, 0, −1⟩.
u = ⟨1, 2, 3⟩, v = ⟨4, 0, −1⟩
1 Multiply by the numbers
2u means 2 times every part of u.
2u = ⟨2 × 1, 2 × 2, 2 × 3⟩
2 times 1 is 2.
2 times 2 is 4.
2 times 3 is 6.
2u = ⟨2, 4, 6⟩
2 Add part by part
Add the matching parts: x with x, y with y, z with z.
2u + v = ⟨2 + 4, 4 + 0, 6 + (−1)⟩
Careful!
The 2 belongs to u only.
v is added as it is.
Careful: not 2(u + v) ✗
So 2u + v = ⟨6, 4, 5⟩.
2u + v = ⟨6, 4, 5⟩
3 Check
Let's check with a picture.
First 2u from the origin.
Then v from its tip.
The arrow from the start to the end is the answer.
Tip to tail: ⟨2, 4, 6⟩ then ⟨4, 0, −1⟩ ends at ⟨6, 4, 5⟩.
Right!
⟨2, 4, 6⟩ + ⟨4, 0, −1⟩ = ⟨6, 4, 5⟩
Let's look back.
First: multiply by the numbers.
Then: add part by part.
Last: check.
Answer
2u + v = ⟨6, 4, 5⟩