Question:
Consider the planes: $P_{1}: 3x + 15y + 21z = 9$, $P_{2}: x - 3y - z = 5$, $P_{3}: 2x + 10y + 14z = 5$. Which of the following is TRUE?
Consider the planes: $P_{1}: 3x + 15y + 21z = 9$, $P_{2}: x - 3y - z = 5$, $P_{3}: 2x + 10y + 14z = 5$. Which of the following is TRUE?
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Planes are parallel if their normal vectors are multiples of each other.
Updated On: Jul 3, 2026
- $P_1$ and $P_2$ are parallel
- $P_1$ and $P_3$ are parallel
- $P_2$ and $P_3$ are parallel
- $P_1, P_2, P_3$ are all parallel
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The Correct Option is B
Solution and Explanation
Step 1: Concept
Two planes $a_1x + b_1y + c_1z = d_1$ and $a_2x + b_2y + c_2z = d_2$ are parallel if $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.
Step 2: Meaning
For $P_1$: $(3, 15, 21)$. For $P_3$: $(2, 10, 14)$.
Step 3: Analysis
Check ratios: $\frac{3}{2} = 1.5$; $\frac{15}{10} = 1.5$; $\frac{21}{14} = 1.5$. The coefficients are proportional.
Step 4: Conclusion
Since the coefficients of $x, y,$ and $z$ are in the same ratio, $P_1$ and $P_3$ are parallel. Final Answer: (B)
Two planes $a_1x + b_1y + c_1z = d_1$ and $a_2x + b_2y + c_2z = d_2$ are parallel if $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.
Step 2: Meaning
For $P_1$: $(3, 15, 21)$. For $P_3$: $(2, 10, 14)$.
Step 3: Analysis
Check ratios: $\frac{3}{2} = 1.5$; $\frac{15}{10} = 1.5$; $\frac{21}{14} = 1.5$. The coefficients are proportional.
Step 4: Conclusion
Since the coefficients of $x, y,$ and $z$ are in the same ratio, $P_1$ and $P_3$ are parallel. Final Answer: (B)
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