$(5, 6, 7) = 5(1, 2, 3) + 0(0, 1, 1)$

$(5, 6, 7) = 2(1, 2, 3) + 1(0, 1, 1)$

Not possible; the two vectors do not span $\mathbb{R}^3$

$(5, 6, 7) = 1(1, 2, 3) + 2(0, 1, 1)$