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

Step by step

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

2 × [1; 0; 3] − 3 × [2; 1; −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.

2 × 1 at the top, and so on down.

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

We work out each times.

= [2; 0; 6] − [6; 3; −3]

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

A minus between the vectors takes away every entry.

= [2 − 6; 0 − 3; 6 − (−3)]

Careful!

Taking away a negative is adding: 6 − (−3) = 6 + 3.

Careful: not 6 − 3 ✗

So the answer is the vector (−4, −3, 9).

= [−4; −3; 9]

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.

= [−4; −3; 9] + [6; 3; −3]

= [−4 + 6; −3 + 3; 9 + (−3)]

= [2; 0; 6]

We get (2, 0, 6): that is 2 × (1, 0, 3), 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

= [−4; −3; 9]