1. Key Concepts and Formulas

• Normal Vector of a Plane: For a plane given by the general equation $Ax + By + Cz = D$, its normal vector, denoted as $\vec{n}$, is $(A, B, C)$. This vector is perpendicular to the plane.
• Condition for Parallel Planes: Two planes are parallel if and only if their normal vectors are parallel. This means that if $\vec{n_1} = (A_1, B_1, C_1)$ and $\vec{n_2} = (A_2, B_2, C_2)$ are the normal vectors of two planes, then they are parallel if their components are proportional: $\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} = k$ for some non-zero constant $k$. If the normal vectors are parallel, the planes are either distinct parallel planes or coincident (the same) planes.
• Distinguishing Parallel and Coincident Planes: If the normal vectors are parallel (i.e., $\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} = k$), then:
  • The planes are distinct parallel planes if $\frac{D_1}{D_2} \neq k$.
  • The planes are coincident planes if $\frac{D_1}{D_2} = k$.

2. Step-by-Step Solution

Step 1: Identify the Equations of the Planes and their Normal Vectors

We are given the equations of three planes. Our first step is to extract their normal vectors and, for ease of comparison, simplify them by dividing by any common factors.

• Plane $P_1$: $3x + 15y + 21z = 9$

  • The normal vector is $\vec{n_1} = (3, 15, 21)$.
  • To simplify, we can divide each component by 3: $\vec{n_1}' = (1, 5, 7)$.
  • The original equation can be simplified by dividing by 3: $x + 5y + 7z = 3$. Here, $D_1' = 3$.

• Plane $P_2$: $x - 3y - z = 5$

  • The normal vector is $\vec{n_2} = (1, -3, -1)$.
  • This vector cannot be simplified further. Here, $D_2 = 5$.

• Plane $P_3$: $2x + 10y + 14z = 5$

  • The normal vector is $\vec{n_3} = (2, 10, 14)$.
  • To simplify, we can divide each component by 2: $\vec{n_3}' = (1, 5, 7)$.
  • The original equation can be simplified by dividing by 2: $x + 5y + 7z = \frac{5}{2}$. Here, $D_3' = \frac{5}{2}$.

Step 2: Compare Normal Vectors for Parallelism

We will now compare the simplified normal vectors for each pair of planes to determine if they are parallel.

• Comparing $P_1$ and $P_2$:

  • We compare $\vec{n_1}' = (1, 5, 7)$ and $\vec{n_2} = (1, -3, -1)$.
  • Let's check the ratios of corresponding components: $\frac{A_1'}{A_2} = \frac{1}{1} = 1$ $\frac{B_1'}{B_2} = \frac{5}{-3} = -\frac{5}{3}$ $\frac{C_1'}{C_2} = \frac{7}{-1} = -7$
  • Since these ratios ($1$, $-\frac{5}{3}$, and $-7$) are not equal, the normal vectors $\vec{n_1}$ and $\vec{n_2}$ are not parallel.
  • Therefore, planes $P_1$ and $P_2$ are not parallel.

• Comparing $P_1$ and $P_3$:

  • We compare $\vec{n_1}' = (1, 5, 7)$ and $\vec{n_3}' = (1, 5, 7)$.
  • We observe that $\vec{n_1}' = \vec{n_3}'$. This means their normal vectors are identical, which implies they are parallel (with a proportionality constant $k=1$).
  • Since their normal vectors are parallel, planes $P_1$ and $P_3$ are parallel.
  • Next, we check if they are distinct parallel planes or coincident planes by comparing their constant terms ($D$ values) from their simplified equations:
    • For $P_1$: $x + 5y + 7z = 3 \implies D_1' = 3$.
    • For $P_3$: $x + 5y + 7z = \frac{5}{2} \implies D_3' = \frac{5}{2}$.
  • Since $D_1' = 3$ and $D_3' = \frac{5}{2}$, and $3 \neq \frac{5}{2}$, the planes $P_1$ and $P_3$ are parallel but distinct.

• Comparing $P_2$ and $P_3$:

  • We compare $\vec{n_2} = (1, -3, -1)$ and $\vec{n_3}' = (1, 5, 7)$.
  • Let's check the ratios of corresponding components: $\frac{A_2}{A_3'} = \frac{1}{1} = 1$ $\frac{B_2}{B_3'} = \frac{-3}{5} = -\frac{3}{5}$ $\frac{C_2}{C_3'} = \frac{-1}{7} = -\frac{1}{7}$
  • Since these ratios ($1$, $-\frac{3}{5}$, and $-\frac{1}{7}$) are not equal, the normal vectors $\vec{n_2}$ and $\vec{n_3}$ are not parallel.
  • Therefore, planes $P_2$ and $P_3$ are not parallel.

Step 3: Evaluate the Options

Based on our detailed analysis:

• (A) $P_1$ and $P_2$ are parallel. (False)
• (B) $P_1$, $P_2$ and $P_3$ all are parallel. (False, as not all pairs are parallel)
• (C) $P_1$ and $P_3$ are parallel. (True)
• (D) $P_2$ and $P_3$ are parallel. (False)

Our mathematical derivation clearly shows that planes $P_1$ and $P_3$ are parallel. This corresponds to option (C).

3. Common Mistakes & Tips

• Always Simplify Normal Vectors: Before comparing, simplify the components of each normal vector by dividing by their greatest common divisor. This makes it easier to spot proportionality.
• Check All Components: For two vectors to be parallel, the ratios of all three corresponding components ($A/A'$, $B/B'$, $C/C'$) must be equal. If even one ratio differs, the vectors are not parallel.
• Distinguish Parallel from Coincident: If the normal vectors are parallel, remember to check the proportionality of the constant terms ($D/D'$) using the scaled equations. This step determines if the planes are distinct or coincident.

4. Summary

To determine if planes are parallel, the key is to examine their normal vectors. If the normal vectors are parallel (i.e., their components are proportional), then the planes are parallel. We systematically extracted the normal vectors for the three given planes, simplified them, and compared them pairwise. Our analysis revealed that the normal vectors for $P_1$ and $P_3$ are proportional (in fact, identical after simplification), indicating that these two planes are parallel. A subsequent check of their constant terms confirmed that they are distinct parallel planes. No other pair of planes had parallel normal vectors. Therefore, the only true statement among the options is that $P_1$ and $P_3$ are parallel.

The final answer is $\boxed{C}$.