Step by step
The question
Question: find 2u + v, 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 Draw tip to tail
Draw the grid.
Draw 2u from the origin.
Put the start of v at the tip of 2u.
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.
2u + v = ⟨4 + 1, 2 + (−3)⟩
Adding a negative is the same as taking away.
+ (−) = −
4 plus 1 is 5.
2 plus (−3) is −1.
So 2u + v = ⟨5, −1⟩.
2u + v = ⟨5, −1⟩
4 Check
Let's check on the grid.
The red arrow ends at (5, −1).
That is right 5 and down 1 from the start: ⟨5, −1⟩.
The same!
tip (5, −1) → ⟨5, −1⟩
Let's look back.
First: multiply by 2.
Then: draw tip to tail.
Then: add the parts.
Last: check.
Answer
2u + v = ⟨5, −1⟩