Step by step
The question
Question: find 2u + 3v, when u = ⟨2, −1⟩ and v = ⟨−1, 3⟩.
u = ⟨2, −1⟩ and v = ⟨−1, 3⟩
1 Multiply by 2
2u is 2 times each part of u.
2u = ⟨2 × 2, 2 × (−1)⟩
So 2u = ⟨4, −2⟩.
2u = ⟨4, −2⟩
2 Multiply by 3
3v is 3 times each part of v.
3v = ⟨3 × (−1), 3 × 3⟩
So 3v = ⟨−3, 9⟩.
3v = ⟨−3, 9⟩
3 Draw tip to tail
Draw the grid.
Draw 2u from the origin.
Put the start of 3v at the tip of 2u.
Tip to tail.
The answer goes from the very start to the very end.
4 Add the parts
Add the across parts, and the up parts.
Each arrow adds its own across and up.
2u + 3v = ⟨4 + (−3), −2 + 9⟩
Adding a negative is the same as taking away.
+ (−) = −
4 plus (−3) is 1.
−2 plus 9 is 7.
So 2u + 3v = ⟨1, 7⟩.
2u + 3v = ⟨1, 7⟩
5 Check
Let's check on the grid.
The red arrow ends at (1, 7).
That is right 1 and up 7 from the start: ⟨1, 7⟩.
The same!
tip (1, 7) → ⟨1, 7⟩
Let's look back.
First: multiply by 2.
Then: multiply by 3.
Then: draw tip to tail.
Then: add the parts.
Last: check.
Answer
2u + 3v = ⟨1, 7⟩