Step by step
The question
Question: find 2u − v, when u = ⟨3, 1⟩ and v = ⟨1, 2⟩.
u = ⟨3, 1⟩ and v = ⟨1, 2⟩
1 Multiply by 2
2u is 2 times each part of u.
2u = ⟨2 × 3, 2 × 1⟩
So 2u = ⟨6, 2⟩.
2u = ⟨6, 2⟩
2 Draw tip to tail
Draw the grid.
Draw 2u from the origin.
Taking away v is adding −v: the arrow turned around.
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 Take away the parts
Take away the across parts, and the up parts.
Taking away an arrow turns it around, so its parts change sign.
2u − v = ⟨6 − 1, 2 − 2⟩
6 minus 1 is 5.
2 minus 2 is 0.
So 2u − v = ⟨5, 0⟩.
2u − v = ⟨5, 0⟩
4 Check
Let's check on the grid.
The red arrow ends at (5, 0).
That is right 5 from the start: ⟨5, 0⟩.
The same!
tip (5, 0) → ⟨5, 0⟩
Let's look back.
First: multiply by 2.
Then: draw tip to tail.
Then: take away the parts.
Last: check.
Answer
2u − v = ⟨5, 0⟩