Show that the distance D between parallel planes ax + by + cz + d1 = 0 and ax + by + cz + d2 = 0 is D = |d1 - d2| / sqrt(a2 + b2 + c2).

To show that the distance D between two parallel planes ax + by + cz + d_1 = 0 and ax + by + cz + d_2 = 0 is given by D = (|d_1 - d_2|)/(sqrt(a^2 + b^2 + c^2)), we can follow these steps: Step 1: Identify the equations of the two parallel planes. Let the first plane be P_1: ax + by + cz + d_1 = 0. Let the second plane be P_2: ax + by + cz + d_2 = 0. Since the planes are parallel, their normal vectors are the same, n = a \\ b \\ c . Step 2: Choose an arbitrary point on one of the planes. Let P_0(x_0, y_0, z_0) be a point on the first plane P_1. Therefore, ax_0 + by_0 + cz_0 + d_1 = 0. From this equation, we can express ax_0 + by_0 + cz_0 = -d_1. Step 3: Use the formula for the distance from a point to a plane. The distance D from a point (x_0, y_0, z_0) to a plane Ax + By + Cz + D_plane = 0 is given by the formula: D = |Ax_0 + By_0 + Cz_0 + D_plane|sqrt(A^2 + B^2 + C^2) We want to find the distance from the point P_0(x_0, y_0, z_0) (which lies on P_1) to the second plane P_2: ax + by + cz + d_2 = 0. In this case, A=a, B=b, C=c, and D_plane=d_2. Step 4: Substitute the point and plane coefficients into the distance formula. Substituting the coordinates of P_0 and the coefficients of P_2 into the distance formula: D = (|ax_0 + by_0 + cz_0 + d_2|)/(sqrt(a^2 + b^2 + c^2)) Step 5: Substitute the expression from Step 2 into the distance formula. We know from Step 2 that ax_0 + by_0 + cz_0 = -d_1. Substitute this into the equation for D: D = (|(-d_1) + d_2|)/(sqrt(a^2 + b^2 + c^2)) D = (|d_2 - d_1|)/(sqrt(a^2 + b^2 + c^2)) Since |d_2 - d_1| is equivalent to |d_1 - d_2|, we can write: D = (|d_1 - d_2|)/(sqrt(a^2 + b^2 + c^2)) This shows the required formula for the distance between two parallel planes. 3 done, 2 left today. You're making progress.