Example 20 - Equation of plane passing through intersection - Equation of plane - Passing Through Intersection Of Planes

Slide7.JPG
Slide8.JPG Slide9.JPG

  1. Class 12
  2. Important Question for exams Class 12
Ask Download

Transcript

Example 20 Find the vector equation of the plane passing through the intersection of the planes 𝑟﷯ . ( 𝑖﷯ + 𝑗﷯ + 𝑘﷯) = 6 and 𝑟﷯ . (2 𝑖﷯ + 3 𝑗﷯ + 4 𝑘﷯)= − 5, and the point (1, 1, 1). The vector equation of a plane passing through the intersection of planes 𝑟﷯. 𝑛1﷯ = d1 and 𝑟﷯. 𝑛2﷯ = d2 and also through the point (x1, y1, z1) is 𝒓﷯.( 𝒏𝟏﷯ + 𝜆 𝒏𝟐﷯) = d1 + 𝜆d2 Given, the plane passes through Equation of plane is 𝑟﷯. 𝑖﷯+ 𝑗﷯+ 𝑘﷯﷯+𝜆(−2 𝑖﷯−3 𝑗﷯−4 𝑘﷯)﷯ = 6 + 𝜆5 𝒓﷯. 𝒊﷯ + 𝒋﷯ + 𝒌﷯﷯−𝜆(𝟐 𝒊﷯+𝟑 𝒋﷯+𝟒 𝒌﷯)﷯ = 6 + 5𝜆 Now to find 𝜆 , put 𝒓﷯ = x 𝒊﷯ + y 𝒋﷯ + z 𝒌﷯ (x 𝑖﷯ + y 𝑗﷯ + z 𝑘﷯). 𝑖﷯+ 𝑗﷯+ 𝑘﷯﷯−𝜆(2 𝑖﷯+3 𝑗﷯+4 𝑘﷯)﷯ = 5𝜆 + 6 (x 𝑖﷯ + y 𝑗﷯ + z 𝑘﷯). 𝑖﷯+ 𝑗﷯+ 𝑘﷯﷯ − 𝜆 (x 𝑖﷯ + y 𝑗﷯ + z 𝑘﷯).(2 𝑖﷯+3 𝑗﷯+4 𝑘﷯) = 5𝜆 + 6 (x × 1) + (y × 1) + (z × 1) − 𝜆 𝑥×2﷯+ 𝑦×3﷯+(𝑧×4)﷯ = 5𝜆 + 6 x + y + z − 𝜆 2𝑥+3𝑦+4𝑧﷯ = 5𝜆 + 6 x + y + z − 2𝜆𝑥 − 3𝜆y − 4𝜆z = 5𝜆 + 6 (1 − 2𝜆)x + (1 − 3𝜆)y + (1 − 4𝜆) z = 5𝜆 + 6 Since the plane passes through (1, 1, 1), Putting (1, 1, 1) in (2) (1 − 2𝜆)x + (1 − 3𝜆)y + (1 − 4𝜆) z = 5𝜆 + 6 (1 −2𝜆) × 1 + (1 − 3𝜆) × 1 + (1 − 4𝜆) × 1 = 5𝜆 + 6 1 −2𝜆 + 1 − 3𝜆 + 1 − 4𝜆= 5𝜆 + 6 3 − 9𝜆 = 5𝜆 + 6 −14𝜆 = 3 ∴ 𝜆 = − 𝟑﷮𝟏𝟒﷯ Putting value of 𝜆 in (1), 𝑟﷯. 𝑖﷯ + 𝑗﷯ + 𝑘﷯﷯− − 3﷮14﷯﷯(2 𝑖﷯+3 𝑗﷯+4 𝑘﷯)﷯= 6 + 5 × − 3﷮14﷯ 𝑟﷯. 𝑖﷯+ 𝑗﷯+ 𝑘﷯﷯+ 3﷮14﷯(2 𝑖﷯+3 𝑗﷯+4 𝑘﷯)﷯= 6 − 15﷮14﷯ 𝑟﷯. 𝑖﷯+ 𝑗﷯ + 𝑘﷯+ 6﷮14﷯ 𝑖﷯+ 9﷮14﷯ 𝑗﷯+ 12﷮14﷯ 𝑘﷯﷯= 69﷮14﷯ 𝑟﷯. 1+ 6﷮14﷯﷯ 𝑖﷯ + 1+ 9﷮14﷯﷯ 𝑗﷯+ 1+ 12﷮14﷯﷯ 𝑘﷯﷯= 69﷮14﷯ 𝑟﷯. 20﷮14﷯ 𝑖﷯ + 23﷮14﷯ 𝑗﷯ + 26﷮14﷯ 𝑘﷯﷯= 69﷮14﷯ 𝑟﷯. 1﷮14﷯(20 𝑖﷯+23 𝑗﷯+26 𝑘﷯)﷯= 69﷮14﷯ 1﷮14﷯ 𝑟﷯. (20 𝑖﷯ + 23 𝑗﷯ + 26 𝑘﷯) = 69﷮14﷯ 𝑟﷯. (20 𝑖﷯ + 23 𝑗﷯ + 26 𝑘﷯) = 69 Therefore, the vector equation of the required plane is 𝒓﷯.(𝟐𝟎 𝒊﷯ + 𝟐𝟑 𝒋﷯ + 𝟐𝟔 𝒌﷯) = 𝟔𝟗

About the Author

CA Maninder Singh's photo - Expert in Practical Accounts, Taxation and Efiling
CA Maninder Singh
CA Maninder Singh is a Chartered Accountant for the past 8 years. He provides courses for Practical Accounts, Taxation and Efiling at teachoo.com .
Jail