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

Step by step

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

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

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

We work out each times.

= [2; 0; 2] + [0; 3; 3] − [1; 1; 0]

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

= [2 + 0 − 1; 0 + 3 − 1; 2 + 3 − 0]

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 (1, 2, 5).

= [1; 2; 5]

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.

= [1; 2; 5] − [0; 3; 3] + [1; 1; 0]

= [1 − 0 + 1; 2 − 3 + 1; 5 − 3 + 0]

= [2; 0; 2]

We get (2, 0, 2): that is 2 × (1, 0, 1), 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; 2; 5]