Question:
Consider the three planes
\(P_1: 3x+15y+21z=9\),
\(P_2: x-3y-z=5\), and
\(P_3: 2x+10y+14z=5\)
Then, which one of the following is true?
Consider the three planes
\(P_1: 3x+15y+21z=9\),
\(P_2: x-3y-z=5\), and
\(P_3: 2x+10y+14z=5\)
Then, which one of the following is true?
\(P_1: 3x+15y+21z=9\),
\(P_2: x-3y-z=5\), and
\(P_3: 2x+10y+14z=5\)
Then, which one of the following is true?
Show Hint
Compare the coefficient ratios of the planes' normals.
Updated On: Nov 4, 2026
- \(P_1\) and \(P_2\) are parallel.
- \(P_1\), \(P_2\) and \(P_3\) all are parallel.
- \(P_1\) and \(P_3\) are parallel.
- \(P_2\) and \(P_3\) are parallel.
Show Solution
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The Correct Option is C
Solution and Explanation
Normals: \(\vec n_1=(3,15,21)=3(1,5,7)\), \(\vec n_2=(1,-3,-1)\), \(\vec n_3=(2,10,14)=2(1,5,7)\).
\(\vec n_1\parallel\vec n_3\) (ratios \(\frac32=\frac{15}{10}=\frac{21}{14}\)), while \(\vec n_2\) is not proportional to either.
Also \(\frac{9}{5}\ne\frac32\), so \(P_1\) and \(P_3\) are distinct parallel planes.
Answer: (3) \(P_1\) and \(P_3\) are parallel.
\(\vec n_1\parallel\vec n_3\) (ratios \(\frac32=\frac{15}{10}=\frac{21}{14}\)), while \(\vec n_2\) is not proportional to either.
Also \(\frac{9}{5}\ne\frac32\), so \(P_1\) and \(P_3\) are distinct parallel planes.
Answer: (3) \(P_1\) and \(P_3\) are parallel.
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