The plane, passing through the points $$(0, -1, 2)$$ and $$(-1, 2, 1)$$ and parallel to the line passing through $$(5, 1, -7)$$ and $$(1, -1, -1)$$, also passes through the point

The plane passes through $$A(0, -1, 2)$$ and $$B(-1, 2, 1)$$.

$$\vec{AB} = (-1, 3, -1)$$

The line passes through $$(5, 1, -7)$$ and $$(1, -1, -1)$$ with direction $$\vec{d} = (-4, -2, 6)$$ or simplified $$(2, 1, -3)$$.

Normal to the plane: $$\vec{n} = \vec{AB} \times \vec{d}$$

$$\vec{n} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -1 & 3 & -1 \\ 2 & 1 & -3 \end{vmatrix}$$

$$= \hat{i}(-9+1) - \hat{j}(3+2) + \hat{k}(-1-6)$$

$$= (-8, -5, -7)$$ or $$(8, 5, 7)$$

Plane equation through $$(0, -1, 2)$$:

$$8(x-0) + 5(y+1) + 7(z-2) = 0$$

$$8x + 5y + 7z = 9$$

Checking which point lies on this plane:

Option 1: $$(-2, 5, 0)$$: $$8(-2) + 5(5) + 7(0) = -16 + 25 = 9$$ ✓

This matches option 1: $$(-2, 5, 0)$$.