Calculate 3u - 2v for u = (1, 2, 0), v = (2, -1, 4)

Step by step

The question

Question: work out 3u − 2v.

u is (1, 2, 0), and v is (2, −1, 4).

We put them in.

So 3u − 2v 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.

3 × 1 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.

A minus between the vectors takes away every entry.

Careful!

Taking away a negative is adding: 6 − (−2) = 6 + 2.

So the answer is the vector (−1, 8, −8).

3 Check

Let's check.

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

The first part must come back.

We get (3, 6, 0): that is 3 × 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.