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