Calculate u + 2v - w for u = (1, 0, 2), v = (0, 1, -1), w = (3, 1, 1)

Step by step

The question

Question: work out u + 2v − w.

u is (1, 0, 2), and v is (0, 1, −1), and w is (3, 1, 1).

We put them in.

So u + 2v − w is this.

A vector in R 3 is a column of 3 numbers.

Numbers times vectors, added together, is called a linear combination.

1 Multiply each vector by its number

Each vector is multiplied by its number: every entry.

2 × 0 at the top, and so on down.

We work out each times.

2 Add matching entries

Now matching entries go together: top with top, then the next, down to the bottom.

Careful!

Every entry gets the same treatment.

Top with top, second with second, down to the bottom.

So the answer is the vector (−2, 1, −1).

3 Check

Let's check.

We take the other parts away from the answer (taking away a minus part is adding it back).

The first part must come back.

We get (1, 0, 2): that is u, the first part.

So the answer is right!

Let's look back.

First: multiply each vector by its number.

Then: add matching entries.

Last: check.