How to check if a vector is in the span of two vectors?

I need to check if (my book writes these as column vectors)





(2, 6, 6) is in the span of (-1, 2, 3) and (3, 4, 2).





So I know what it's asking. It's asking if some linear combination of the last two vectors produces the 1st vector.|||Check the determinant of matrix


2 -1 3


6 2 4


6 3 2


If it is =0 it means that (2,6,6) is a linear combination of the last two vectors.


If not all three vectors are independent.|||Let r, s be some real numbers. For [2, 6, 6] to be a linear combination of the other two vectors: there must be some scalar multiple of the other two that gives us the [2, 6, 6] vector.





[2, 6, 6] = r [-1, 2, 3] + s [3, 4, 2]


Then:





2 = -r + 3s


6 = 2r + 4s


6 = 3r + 2s





Now to solve for r and s. If we can find some values for them that works, then it is in the span of the two vectors.





Add 2 of the first equation to the 2nd equation:


6 + 4 = (2r + 4s) + (-2s + 6s)


10 = 10s


s = 1





Plug s = 1 into one of the original equations.





6 = 3r + 2s


6 = 3r + 2(1)


6 - 2 = 3r


4 = 3r


r = 4/3





Plug these values into the first equation:


2 = -r + 3s


2 = (-4/3) + 3


-1 = -4/3





That is obviously not true. Thus, not in the span.