The acute angle between the planes P1 and P2, when P1 and P2 are the planes passing through the intersection of the planes 5x + 8y + 13z – 29 = 0 and 8x – 7y + z – 20 = 0 and the points (2, 1, 3) and (0, 1, 2), respectively, is
\(\frac{π}{3}\)
\(\frac{π}{4}\)
\(\frac{π}{6}\)
\(\frac{π}{12}\)
The Correct Option is A
Solution and Explanation
The correct answer is (A) : \(\frac{π}{3}\)
Family of Plane’s equation can be given by
(5 + 8λ)x + (8 – 7λ)y + (13 + λ) z – (29 + 20λ) = 0
P1 passes through (2, 1, 3)
⇒ (10 + 16λ) + (8 – 7λ) + (39 + 3λ) – (29 + 20λ) = 0
⇒ –8λ + 28 = 0
\(⇒ λ =\frac{7}{2}\)
d.r,s of normal to P1
\(⟨33, \frac{−33}{2} ,\frac{33}{2}⟩ or ⟨1, \frac{−1}{2}, \frac{1}{2}⟩\)
P2 passes through (0, 1, 2)
8 – 7λ + 26 + 2λ – (29 + 20λ) = 0
⇒ 5 – 25λ = 0
\(⇒ λ = \frac{1}{5}\)
d.r, s of normal to P2
\(⟨ \frac{33}{5} ,\frac{33}{5}, \frac{66}{5}⟩ or ⟨1, 1, 2⟩\)
Angle between normals
\(= \frac{( \hat{i} -\frac{1}{2}\hat{j} + \frac{1}{2}\hat{k} ) . ( \hat{i} + \hat{j} + 2\hat{k} )}{ \frac{\sqrt{3}}{2} \hspace{1cm}\sqrt6}\)
\(cosθ = \frac{1 -\frac{1}{2} + 1}{3} = \frac{1}{2}\)
\(θ = \frac{π}{3}\)
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Concepts Used:
Three Dimensional Geometry
Mathematically, Geometry is one of the most important topics. The concepts of Geometry are derived w.r.t. the planes. So, Geometry is divided into three major categories based on its dimensions which are one-dimensional geometry, two-dimensional geometry, and three-dimensional geometry.
Direction Cosines and Direction Ratios of Line:
Consider a line L that is passing through the three-dimensional plane. Now, x,y and z are the axes of the plane and α,β, and γ are the three angles the line makes with these axes. These are commonly known as the direction angles of the plane. So, appropriately, we can say that cosα, cosβ, and cosγ are the direction cosines of the given line L.
