Step by step
The question
Question: work out (5, 2, 3) − (2, 2, 1).
[5; 2; 3] − [2; 2; 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
1 Add matching entries
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 − 2; 2 − 2; 3 − 1]
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 (3, 0, 2).
= [3; 0; 2]
2 Check
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.
= [3; 0; 2] + [2; 2; 1]
= [3 + 2; 0 + 2; 2 + 1]
= [5; 2; 3]
We get (5, 2, 3): that is (5, 2, 3), the first part.
So the answer is right!
Let's look back.
First: add matching entries.
Last: check.
Answer
= [3; 0; 2]