1. The problem statement, all variables and given/known data I'm trying to find the coordinates of a vector [ 7 4 1 ] relative to these three vectors: [ -1 2 -9 ] , [ 1 5 1 ] , [ 47 -8 -7 ] 2. Relevant equations none 3. The attempt at a solution Do I get all these vectors into an augumented matrix? like: [ -1 1 47 | 7 ] [ 2 5 -8 | 4 ] [ -9 1 -7 | 1 ] and then row reduce completely? I got very awkward numbers and just wanted to make sure I'm using the correct method.
I take it that "relative to these three vectors" means write it as a linear combination of those vectors. What you want are three numbers, a, b, c such that a<-1, 2, 9>+ b<1, 5, 1>+ c<47, -8, -7>= <7, 4, 1>. Looking at x, y, and z components separately that means -a+ b+ 47c= 7, 2a+ 5b- 8c= 4, 9a+ b- 7c= 1 which gives exactly the augmented matrix you have. Are you sure about that "47"? That number looks out of place.
yes that 47 must be there because it was part of another question where I had to find a third vector orthogonal to the other two (involving the cross product) and I'm confident about that. If you say that my setup is right, then my more awkward numbers must be right. I got a = -4/43 b = 28/27 c = 145/1161