Answer:
[tex](u + v) + w = u + (v + w)\\(0,4)=(0,4)[/tex]
Step-by-step explanation:
We have:
[tex]u=(1,2), v=(3,-4), w=(-4,6)[/tex]
And we have to prove: [tex](u + v) + w = u + (v + w)[/tex]
First we have to solve the parentheses.
Observation:
If you have two vectors: [tex]A=(a,b)[/tex] and [tex]B=(c,d)[/tex]
[tex]A+B=(a,b)+(c,d)=(a+c,b+d)[/tex]
First we are going to calculate: [tex](u + v) + w[/tex]
[tex](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)[/tex]
Now we have to calculate: [tex]u + (v + w)[/tex]
[tex]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)[/tex]
Then,
[tex](u + v) + w = u + (v + w)\\(0,4)=(0,4)[/tex]
It's verified.
