u = (3, −2) and v = (−1, 4) as column vectors. Find 2u − v. Step by Step

Step by step

The question

Question: find 2u − v, when u = ⟨3, −2⟩ and v = ⟨−1, 4⟩.

u = ⟨3, −2⟩ and v = ⟨−1, 4⟩

1 Multiply by 2

2u is 2 times each part of u.

2u = ⟨2 × 3, 2 × (−2)⟩

2 times 3 is 6.

2 times (−2) is −4.

2 × 3 = 6

2 × (−2) = −4

So 2u = ⟨6, −4⟩.

2u = ⟨6, −4⟩

2 Draw tip to tail

Draw the grid.

Draw 2u from the origin.

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

The tail of an arrow is its start.

The tip is its end, with the arrowhead.

Put the tail of −v at the tip of 2u: tip to tail.

The answer arrow, in red, goes from the very start to the very end.

It is one move that does both moves.

3 Take away the parts

We take away the across parts, and the up parts.

Why?

Taking away an arrow turns it around.

Then its parts have the other sign.

2u − v = ⟨6 − (−1), −4 − 4⟩

Taking away a negative is the same as adding.

− (−) = +

Careful!

Minus a negative is plus.

6 minus (−1) is 6 plus 1.

Careful: not 6 − (−1) = 5 ✗

6 minus (−1) is 7.

−4 minus 4 is −8.

6 − (−1) = 7

−4 − 4 = −8

So 2u − v = ⟨7, −8⟩.

2u − v = ⟨7, −8⟩

4 Check

Let's check on the grid.

The red answer arrow ends at (7, −8).

That is right 7 and down 8 from the start: ⟨7, −8⟩.

The same!

tip (7, −8) → ⟨7, −8⟩

Let's look back.

First: multiply by 2.

Then: draw tip to tail.

Then: take away the parts.

Last: check.

Answer