u = <4, 2>, v = <1, -1>, find u + 2v

Step by step

The question

Question: find u + 2v, when u = ⟨4, 2⟩ and v = ⟨1, −1⟩.

1 Multiply by 2

2v is 2 times each part of v.

So 2v = ⟨2, −2⟩.

2 Draw tip to tail

Draw the grid.

Draw u from the origin.

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 Add the parts

Add the across parts, and the up parts.

Each arrow adds its own across and up.

Adding a negative is the same as taking away.

4 plus 2 is 6.

2 plus (−2) is 0.

So u + 2v = ⟨6, 0⟩.

4 Check

Let's check on the grid.

The red arrow ends at (6, 0).

That is right 6 from the start: ⟨6, 0⟩.

The same!

Let's look back.

First: multiply by 2.

Then: draw tip to tail.

Then: add the parts.

Last: check.