Step by step
The question
Question: find 3u + v, when u = ⟨−1, 2⟩ and v = ⟨2, 1⟩.
u = ⟨−1, 2⟩ and v = ⟨2, 1⟩
1 Multiply by 3
3u is 3 times each part of u.
3u = ⟨3 × (−1), 3 × 2⟩
So 3u = ⟨−3, 6⟩.
3u = ⟨−3, 6⟩
2 Draw tip to tail
Draw the grid.
Draw 3u from the origin.
Put the start of v at the tip of 3u.
Tip to tail.
The answer goes from the very start to the very end.
3 Add the parts
Add the across parts, and the up parts.
Each arrow adds its own across and up.
3u + v = ⟨−3 + 2, 6 + 1⟩
−3 plus 2 is −1.
6 plus 1 is 7.
So 3u + v = ⟨−1, 7⟩.
3u + v = ⟨−1, 7⟩
4 Check
Let's check on the grid.
The red arrow ends at (−1, 7).
That is left 1 and up 7 from the start: ⟨−1, 7⟩.
The same!
tip (−1, 7) → ⟨−1, 7⟩
Let's look back.
First: multiply by 3.
Then: draw tip to tail.
Then: add the parts.
Last: check.
Answer
3u + v = ⟨−1, 7⟩