Check sibling questions

Misc 6 - If lines are perpendicular find value of k. x-1/-3 = y-2/2k

Misc 6 - Chapter 11 Class 12 Three Dimensional Geometry - Part 2

This video is only available for Teachoo black users

Get live Maths 1-on-1 Classs - Class 6 to 12


Transcript

Misc 6 If the lines (𝑥 − 1)/( − 3) = (𝑦 − 2)/2𝑘 = (𝑧 − 3)/2 and (𝑥 − 1)/3𝑘 = (𝑦 − 1)/1 = (𝑧 − 6)/( − 5) are perpendicular, find the value of k. Two lines (𝑥 − 𝑥1)/( 𝑎1) = (𝑦 −𝑦1)/𝑏1 = (𝑧 − 𝑧1)/𝑐1 and (𝑥 − 𝑥2)/( 𝑎2) = (𝑦 − 𝑦2)/𝑏2 = (𝑧 − 𝑧2)/𝑐2 are perpendicular to each other if 𝒂𝟏 𝒂𝟐 + 𝒃𝟏 𝒃𝟐 + 𝒄𝟏 𝒄𝟐 = 0 (𝒙 − 𝟏)/( − 𝟑) = (𝒚 − 𝟐)/( 𝟐𝒌) = (𝒛 − 𝟑)/( 𝟐) Comparing with (𝑥 − 𝑥1)/( 𝑎1) = (𝑦− 𝑦1)/( 𝑏1) = (𝑧− 𝑧1)/( 𝑐1) x1 = 1, y1 = 2, z1 = 3 & 𝒂𝟏 = −3, b1 = 2k c1 = 2 (𝒙 − 𝟏)/( 𝟑𝒌) = (𝒚 − 𝟏)/( 𝟏) = (𝒛 − 𝟔)/( − 𝟓) Comparing with (𝑥 − 𝑥2)/( 𝑎2) = (𝑦 − 𝑦2)/( 𝑏2) = (𝑧 − 𝑧2)/( 𝑐2), 𝑥2 = 1, y2 = 1, z2 = 6 & 𝒂𝟐 = 3k, b2 = 1, c2 = −5, Since the two lines are perpendicular, 𝑎1 𝑎2 + 𝑏1 𝑏2 + c1𝑐2 = 0 (−3 × 3k) + (2k × 1) + (2 × − 5) = 0 − 9k + 2k − 10 = 0 − 7k = 10 k = (−𝟏𝟎)/𝟕 Therefore, k = (−10)/7 Misc 7 Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane 𝑟 ⃗.(𝑖 ̂ + 2𝑗 ̂ − 5𝑘 ̂) + 9 = 0 . The vector equation of a line passing through a point with position vector 𝑎 ⃗ and parallel to vector 𝑏 ⃗ is 𝒓 ⃗ = 𝒂 ⃗ + 𝜆𝒃 ⃗ Given, the line passes (1, 2, 3) So, 𝒂 ⃗ = 1𝒊 ̂ + 2𝒋 ̂ + 3𝒌 ̂ Finding normal of plane 𝑟 ⃗.(𝑖 ̂ + 2𝑗 ̂ − 5𝑘 ̂) + 9 = 0 𝑟 ⃗. (𝑖 ̂ + 2𝑗 ̂ − 5𝑘 ̂) = − 9 −𝑟 ⃗. (𝑖 ̂ + 2𝑗 ̂ − 5𝑘 ̂) = 9 𝑟 ⃗. (−1𝑖 ̂ − 2𝑗 ̂ + 5𝑘 ̂) = 9 Comparing with 𝑟 ⃗. 𝑛 ⃗ = d, 𝒏 ⃗ = −𝑖 ̂ − 2𝑗 ̂ + 5𝑘 ̂ Since line is perpendicular to plane, the line will be parallel to the normal of the plane ∴ 𝒃 ⃗ = 𝒏 ⃗ = −1𝑖 ̂ − 2𝑖 ̂ + 5𝑘 ̂ Hence, 𝑟 ⃗ = (1𝑖 ̂ + 2𝑖 ̂ + 3𝑘 ̂) + 𝜆(−1𝑖 ̂ − 2𝑖 ̂ + 5𝑘 ̂) 𝑟 ⃗ = (1𝑖 ̂ + 2𝑖 ̂ + 3𝑘 ̂) − 𝜆(1𝑖 ̂ + 2𝑖 ̂ − 5𝑘 ̂) 𝒓 ⃗ = (𝒊 ̂ + 2𝒋 ̂ + 3𝒌 ̂) + k (𝒊 ̂ + 2𝒋 ̂ − 5𝒌 ̂)

Ask a doubt
Davneet Singh's photo - Co-founder, Teachoo

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.