Step by step
The question
Question: find 2v, when v = ⟨3, −1⟩.
v = ⟨3, −1⟩
1 Draw the arrows
Draw the grid.
Draw v = ⟨3, −1⟩ from the origin.
2v means 2 times v: 2 arrows v, tip to tail.
Together they make one arrow, 2 times as long.
That is 2v.
2 Multiply each part
2v means: multiply each part of v by 2.
2v = ⟨2 × 3, 2 × (−1)⟩
Careful!
Both parts are multiplied, not only the first one.
The arrow gets longer across and up.
Careful: not ⟨6, −1⟩ ✗
2 times 3 is 6.
2 times (−1) is −2.
So 2v = ⟨6, −2⟩.
2v = ⟨6, −2⟩
3 Check
Let's check on the grid.
The red arrow ends at (6, −2).
That is right 6 and down 2 from the start: ⟨6, −2⟩.
The same!
tip (6, −2) → ⟨6, −2⟩
Let's look back.
First: draw the arrows.
Then: multiply each part.
Last: check.
Answer
2v = ⟨6, −2⟩