Misc 10 - Find coordinates point where line crosses YZ-plane - Point with Lines and Planes

Slide31.JPG
Slide32.JPG

Slide33.JPG Slide34.JPG Slide35.JPG

  1. Chapter 11 Class 12 Three Dimensional Geometry
  2. Serial order wise
Ask Download

Transcript

Misc 10 (Method 1) Find the coordinates of the point where the line through (5, 1, 6) and (3, 4,1) crosses the YZ-plane. The equation of a line passing through two points with position vectors 𝑎﷯ & 𝑏﷯ is 𝒓﷯ = 𝒂﷯ + 𝜆( 𝒃﷯ − 𝒂﷯) Given, the line passes through ( 𝑏﷯ − 𝑎﷯) = (3 𝑖﷯ + 4 𝑗﷯ + 1 𝑘﷯) − (5 𝑖﷯ + 1 𝑗﷯ + 6 𝑘﷯) = (3 −5) 𝑖﷯ + (4 − 1) 𝑗﷯ + (1 − 6) 𝑘﷯ = −2 𝑖﷯ + 3 𝑗﷯ − 5 𝑘﷯ ∴ 𝒓﷯ = (5 𝒊﷯ + 𝒋﷯ + 6 𝒌﷯) + 𝜆 (−2 𝒊﷯ + 3 𝒋﷯ − 5 𝒌﷯) Let the coordinates of the point where the line crosses the YZ plane be (0, y, z) So, 𝒓﷯ = 0 𝒊﷯ + y 𝒋﷯ + z 𝒌﷯ Since point lies in line, it will satisfy its equation, Putting (2) in (1) 0 𝑖﷯ + y 𝑗﷯ + z 𝑘﷯ = 5 𝑖﷯ + 𝑗﷯ + 6 𝑘﷯ −2𝜆 𝑖﷯ + 3𝜆 𝑗﷯ − 5𝜆 𝑘﷯ 0 𝑖﷯ + y 𝑗﷯ + z 𝑘﷯ = (5 −2𝜆) 𝑖﷯ + (1 + 3𝜆) 𝑗﷯ + (6 − 5𝜆) 𝑘﷯ Two vectors are equal if their corresponding components are equal So, Solving 0 = 5 − 2𝜆 5 = 2𝜆 ∴ 𝜆 = 𝟓﷮𝟐﷯ Now, y = 1 + 3𝜆 = 1 + 3 × 5﷮2﷯ = 1 + 15﷮2﷯ = 17﷮2﷯ & z = 6 − 5𝜆 = 6 − 5 × 5﷮2﷯ = 6 − 25﷮2﷯ = −13﷮2﷯ Therefore, the coordinates of the required point is 𝟎, 𝟏𝟕﷮𝟐﷯, −𝟏𝟑﷮𝟐﷯﷯. Misc 10 (Method 2) Find the coordinates of the point where the line through (5, 1, 6) and (3, 4,1) crosses the YZ-plane. The equation of a line passing through two points A( 𝑥﷮1﷯, 𝑦﷮1﷯, 𝑧﷮1﷯) and B( 𝑥﷮2﷯, 𝑦﷮2﷯, 𝑧﷮2﷯) is 𝒙 − 𝒙﷮𝟏﷯﷮ 𝒙﷮𝟐﷯ − 𝒙﷮𝟏﷯﷯ = 𝒚 − 𝒚﷮𝟏﷯﷮ 𝒚﷮𝟐﷯ − 𝒚﷮𝟏﷯﷯ = 𝒛 − 𝒛﷮𝟏﷯﷮ 𝒛﷮𝟐﷯ − 𝒛﷮𝟏﷯﷯ Given the line passes through the points So, the equation of line is 𝑥 − 5﷮3 − 5﷯ = 𝑦 − 1﷮4 − 1﷯ = 𝑧 − 6﷮11 − 6﷯ 𝒙 − 𝟓﷮−𝟐﷯ = 𝒚 − 𝟏﷮𝟑﷯ = 𝒛 − 𝟔﷮−𝟓﷯ = k So, Since the line crosses the YZ plane at (0, y, z), x = 0 −2k + 5 = 0 2k = 5 k = 𝟓﷮𝟐﷯ So, x = 2k + 5 = −2 × 5﷮2﷯ + 5 = − 10﷮2﷯ + 5 = 0 y = 3k + 1 = 3 × 5﷮2﷯ + 1 = 15﷮2﷯ + 1 = 17﷮2﷯ & z = −5k + 6 = −5 × 5﷮2﷯ + 6 = −25﷮2﷯ + 6 = −13﷮2﷯ Therefore, the coordinates of the required point are 𝟎, 𝟏𝟕﷮𝟐﷯, −𝟏𝟑﷮𝟐﷯﷯.

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.