Calculate 3(2, 1, -2) - 2(3, 0, -1) + (0, 4, 1) Step by Step

Step by step

Question: work out 3(2, 1, −2) − 2(3, 0, −1) + (0, 4, 1).

3 × [2; 1; −2] − 2 × [3; 0; −1] + [0; 4; 1]

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

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

a vector in R³: a column of 3 numbers

linear combination: numbers × vectors, added

Each vector is multiplied by its number: every entry.

3 × 2 at the top, and so on down.

= [3 × 2; 3 × 1; 3 × (−2)] − [2 × 3; 2 × 0; 2 × (−1)] + [0; 4; 1]

We work out each times.

= [6; 3; −6] − [6; 0; −2] + [0; 4; 1]

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

= [6 − 6 + 0; 3 − 0 + 4; −6 − (−2) + 1]

Careful!

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

Careful: not −6 − 2 ✗

So the answer is the vector (0, 7, −3).

= [0; 7; −3]

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.

= [0; 7; −3] + [6; 0; −2] − [0; 4; 1]

= [0 + 6 − 0; 7 + 0 − 4; −3 + (−2) − 1]

= [6; 3; −6]

We get (6, 3, −6): that is 3 × (2, 1, −2), 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.

Answer

= [0; 7; −3]