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

Step by step

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

4 × [1; 1; 0] − 2 × [0; 3; 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.

4 × 1 at the top, and so on down.

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

We work out each times.

= [4; 4; 0] − [0; 6; 2]

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

A minus between the vectors takes away every entry.

= [4 − 0; 4 − 6; 0 − 2]

Careful!

Every entry gets the same treatment.

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

Careful: every entry: top with top, down to the bottom

So the answer is the vector (4, −2, −2).

= [4; −2; −2]

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; −2; −2] + [0; 6; 2]

= [4 + 0; −2 + 6; −2 + 2]

= [4; 4; 0]

We get (4, 4, 0): that is 4 × (1, 1, 0), 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; −2; −2]