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NCERT Class XII Mathematics Chapter - - Solutions

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Question 49 - 54: In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.
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Question : 50 of 78
Marks: +1, -0
7x + 5y + 6z + 30 = 0 and 3x – y – 10z + 4 = 0
Solution:  
The given planes are
7x + 5y + 6z + 30 = 0 ...(1)
and 3x – y – 10z + 4 = 0 ...(2)
These are neither parallel [7351610]\left[\because \frac{7}{3} \neq \frac{5}{-1} \neq \frac{6}{-10}\right]
nor perpendicular [∵ (7) (3) + (5) (–1) + 6 (– 10) ≠ 0]
If ‘θ’ be the angle between (1) and (2), then
cosθ=(7)(3)+(5)(1)+(6)(10)49+25+369+1+100=25\cos \theta = \left| \frac{(7)(3)+(5)(-1)+(6)(-10)}{\sqrt{49+25+36}\sqrt{9+1+100}} \right| = \left| -\frac{2}{5} \right|
Hence θ=cos1(25)\theta = \cos^{-1}\left(\frac{2}{5}\right)
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