Question

Given u = 〈1,2〉, v = 〈3, −4〉, and w = 〈−4,6〉, show that (u + v) + w = u + (v + w).

Answer 1

`(u + v) + w = u + (v + w)\\(0,4)=(0,4)`

Explanation:

We have:

`u=(1,2), v=(3,-4), w=(-4,6)`

And we have to prove:
`(u + v) + w = u + (v + w)`

First we have to solve the parentheses.

Observation:

If you have two vectors:
`A=(a,b)` and
`B=(c,d)`

`A+B=(a,b)+(c,d)=(a+c,b+d)`

First we are going to calculate:
`(u + v) + w`

`(u + v) + w=((1,2)+(3,-4))+(-4,6)\\(u + v) + w=(1+3,2+(-4))+(-4,6)\\(u + v) + w=(4,-2)+(-4,6)\\(u + v) + w=(4+(-4),(-2)+6)\\(u + v) + w=(0,4)`

Now we have to calculate:
`u + (v + w)`

`u + (v + w)=(1,2)+((3,-4)+(-4,6))\\u + (v + w)=(1,2)+(3+(-4),(-4)+6)\\u + (v + w)=(1,2)+(-1,2)\\u + (v + w)=(1+(-1),2+2)\\u + (v + w)=(0,4)`

Then,

`(u + v) + w = u + (v + w)\\(0,4)=(0,4)`

It's verified.