Calculate 5(1, 2) - 2(3, -4) Step by Step

Step by step

Question: work out 5(1, 2) − 2(3, −4).

5 × [1; 2] − 2 × [3; −4]

A vector in R 2 is a column of 2 numbers.

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

a vector in R²: a column of 2 numbers

linear combination: numbers × vectors, added

Each vector is multiplied by its number: every entry.

5 × 1 at the top, and so on down.

= [5 × 1; 5 × 2] − [2 × 3; 2 × (−4)]

We work out each times.

= [5; 10] − [6; −8]

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

A minus between the vectors takes away every entry.

= [5 − 6; 10 − (−8)]

Careful!

Taking away a negative is adding: 10 − (−8) = 10 + 8.

Careful: not 10 − 8 ✗

So the answer is the vector (−1, 18).

= [−1; 18]

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.

= [−1; 18] + [6; −8]

= [−1 + 6; 18 + (−8)]

= [5; 10]

We get (5, 10): that is 5 × (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

= [−1; 18]