Question

Span of two vectors

Answer 1

Find a, b, e, and f such that the vectors span the plane 3x+6y-z=0.

\[\left(\begin{matrix}-3 \\ a\\ b \end{matrix}\right)\] and \[\left(\begin{matrix}1 \\ e\\f \end{matrix}\right)\]

Answer 2

I tried constructing a matrix and then row reducing such that the 3,3 element was 3x+6y-z and since it is equal to zero that implies that the 3,1 and 3,2 entries need to be equal to zero as well since it is in those cases that we force the plane to be equal to zero in the first place. This yield linear equations, so when I select a value for a variable for each equation and then get the output, I use those in the original matrix to see if it row reduces to having zeros on the bottom row and 2x+6y-z in the 3,3 position. This doesn't seem to work out though. I'm not sure how else to go about solving the problem.

Answer 3

3x+6y-z=0