I have done some progress here.

What we have to get to know is if

u=(1,1,1) , v(4,5,-3) and w = (7,8,1)

are located in the same plane

the coordinates are located in some basis in the room.

1. Lambda (1,1,1)e + Mu(4,5-3)e + Upsilon(7,8,0)e

= (Lambda,Lambda,Lambda)e + (4Mu,5mu,-3Mu)e + (7Upsilon,8Upsilon,0Upsilon)e =

(0,0,0)e

<-> (Lambda + 4Mu + 7Upsilon , Lambda + 5Mu + 8 Upsilon , Lambda - 3 Mu + 0)e =(0,0,0)e

2. This gives us an equation system looking like this:

Lambda + 4mu + 7 upsilon = 0

Lambda + 5mu + 8upsilon = 0

Lambda- 3 Mu = 0

Lambda = 3mu

And substituting it for the lambdas in the above equations gives us:

3mu + 4mu + 7upsilon = 0

3mu + 5mu + 8 upsilon = 0

Lambda = 3mu

<->

7mu + 7upsilon = 0

8mu + 8upsilon = 0

Lambda = 3 mu

I am quite confused on what kind of conlusion I should make of this

Are they in the same plane or not?

How can I show it.

Thanks.