u = <1, 2>, v = <2, -1>, find u

Step by step

The question

Question: find u − 2v, when u = ⟨1, 2⟩ and v = ⟨2, −1⟩.

u = ⟨1, 2⟩ and v = ⟨2, −1⟩

1 Multiply by 2

2v is 2 times each part of v.

2v = ⟨2 × 2, 2 × (−1)⟩

So 2v = ⟨4, −2⟩.

2v = ⟨4, −2⟩

2 Draw tip to tail

Draw the grid.

Draw u from the origin.

Taking away 2v is adding −2v: the arrow turned around.

Put the start of −2v at the tip of u.

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.

u − 2v = ⟨1 − 4, 2 − (−2)⟩

Taking away a negative is the same as adding.

− (−) = +

Careful!

Minus a negative is plus.

2 minus (−2) is 2 plus 2.

Careful: not 2 − (−2) = 0 ✗

1 minus 4 is −3.

2 minus (−2) is 4.

So u − 2v = ⟨−3, 4⟩.

u − 2v = ⟨−3, 4⟩

4 Check

Let's check on the grid.

The red arrow ends at (−3, 4).

That is left 3 and up 4 from the start: ⟨−3, 4⟩.

The same!

tip (−3, 4) → ⟨−3, 4⟩

Let's look back.

First: multiply by 2.

Then: draw tip to tail.

Then: take away the parts.

Last: check.

Answer