Last updated at Dec. 8, 2016 by Teachoo

Transcript

Misc 11 (Method 1) Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the ZX-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 ZX plane be (x, 0, z) So, 𝒓 = x 𝒊 + 0 𝒋 + z 𝒌 Since point lies in line, it will satisfy its equation, Putting (2) in (1) x 𝑖 + 0 𝑗 + z 𝑘 = 5 𝑖 + 𝑗 + 6 𝑘 −2𝜆 𝑖 + 3𝜆 𝑗 − 5𝜆 𝑘 x 𝑖 + 0 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 = 1 + 3𝜆 3𝜆 = −1 ∴ 𝜆 = −𝟏𝟑 Now, x = 5 − 2𝜆 = 5 − 2 × −13 = 5 + 23 = 1713 z = 6 − 5𝜆 = 6 − 5 × −13 = 6 + 53 = 233 Therefore, the coordinate of the required point are 𝟏𝟕𝟑,𝟎, 𝟐𝟑𝟑 Misc 11 (Method 2) Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the ZX-plane. The equation of a line passing through two points A( 𝑥1, 𝑦1, 𝑧1) and B( 𝑥2, 𝑦2, 𝑧2) is 𝒙 − 𝒙𝟏 𝒙𝟐 − 𝒙𝟏 = 𝒚 − 𝒚𝟏 𝒚𝟐 − 𝒚𝟏 = 𝒛 − 𝒛𝟏 𝒛𝟐 − 𝒛𝟏 Given the line passes through the points So, the equation of line is 𝑥 − 53 − 5 = 𝑦 − 14 − 1 = 𝑧 − 611 − 6 𝒙 − 𝟓−𝟐 = 𝒚 − 𝟏𝟑 = 𝒛 − 𝟔−𝟓 = k So, Since the line crosses the ZX plane at (x, 0, z), y = 0 3k + 1 = 0 3k = −1 ∴ k = −𝟏𝟑 So, x = –2k + 5 = −2 × −13 + 5 = 23 + 5 = 173 y = 0 & z = −5k + 6 = −5 × −13 + 6 = 53 + 6 = 233 therefore, the coordinate of the required point are 𝟏𝟕𝟑,𝟎, 𝟐𝟑𝟑

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.