1. The problem statement, all variables and given/known data Given the vectors a = (5,2,-1), b = (3,2,1), c = (1,2,3), b' = (1,1,0), c' = (3,-3,-2) We assume that the vector a is a linear combination of the vectors b and c and b' and c' respectively, so that: a = xb + yc = x'b' + y'c' a) Determine the factors x and y through construction of different scalar products from the given equation. b) Repeat the calculation for the vectors b' and c' to obtain x' and y'. What do you notice? c) How can you verify the conditions given at the beginning? 2. Relevant equations Anything about basic vector operations I guess. 3. The attempt at a solution I think I was able to solve a) and b), but my question is that I have not been using scalar product, but instead a simple system of linear equations: a) a = xb + yc = (3x,2x,x) + (y,2y,3y) = (3x + y, 2x + 2y, x + 3y) From there, I determine x and y using the fact that I know the components of the vector a: 3x + y = 5 2x + 2y = 2 x = 2 and y = -1 b) a = x'b' + y'c' = (x',x',0) + (3y',-3y',-2y') = (x' + 3y', x' - 3y', -2y') Using the same method as a), I obtain: x' = 7/3 and y' = 1/2 Not only did I not use scalar product (or did I? since multiplying a vector by a constant is considered a scalar product, right?), but I also notice nothing really :) I would be grateful to read your suggestions. Thank you in advance, I appreciate your help. J.