Chapter 11 Class 12 Three Dimensional Geometry
Ex 11.1, 2
Example, 6 Important
Example, 7
Example 10 Important
Ex 11.2, 5 Important
Ex 11.2, 9 (i) Important
Ex 11.2, 10 Important
Ex 11.2, 12 Important
Ex 11.2, 13 Important
Ex 11.2, 15 Important
Question 10 Important Deleted for CBSE Board 2024 Exams
Question 11 Important Deleted for CBSE Board 2024 Exams
Question 13 Important Deleted for CBSE Board 2024 Exams
Question 14 Deleted for CBSE Board 2024 Exams
Question 15 Important Deleted for CBSE Board 2024 Exams
Question 4 (a) Important Deleted for CBSE Board 2024 Exams
Question 11 Important Deleted for CBSE Board 2024 Exams
Question 12 Important Deleted for CBSE Board 2024 Exams
Question 14 (a) Important Deleted for CBSE Board 2024 Exams
Question 17 Important Deleted for CBSE Board 2024 Exams
Question 19 Important Deleted for CBSE Board 2024 Exams
Question 20 Important Deleted for CBSE Board 2024 Exams
Misc 3 Important
Misc 4 Important
Question 10 Important Deleted for CBSE Board 2024 Exams
Question 14 Important Deleted for CBSE Board 2024 Exams
Misc 5 Important You are here
Question 16 Important Deleted for CBSE Board 2024 Exams
Chapter 11 Class 12 Three Dimensional Geometry
Last updated at April 16, 2024 by Teachoo
Misc 5 (Method 1) Find the vector equation of the line passing through the point (1, 2, –4) and perpendicular to the two lines: (𝑥 − 8)/3 = (𝑦 + 19)/(−16) = (𝑧 − 10)/7 and (𝑥 − 15)/3 = (𝑦 − 29)/8 = (𝑧 − 5)/(−5) The vector equation of a line passing through a point with position vector 𝑎 ⃗ and parallel to a vector 𝑏 ⃗ is 𝒓 ⃗ = 𝒂 ⃗ + 𝜆𝒃 ⃗ The line passes through (1, 2, −4) So, 𝒂 ⃗ = 1𝒊 ̂ + 2𝒋 ̂ − 4𝒌 ̂ Given, line is perpendicular to both lines ∴ 𝑏 ⃗ is perpendicular to both lines We know that 𝑥 ⃗ × 𝑦 ⃗ is perpendicular to both 𝑥 ⃗ & 𝑦 ⃗ So, 𝒃 ⃗ is cross product of both lines (𝑥 − 8)/3 = (𝑦 + 19)/(−16) = (𝑧 − 10)/7 and (𝑥 − 15)/3 = (𝑦 − 29)/8 = (𝑧 − 5)/(−5) Required normal = |■8(𝑖 ̂&𝑗 ̂&𝑘 ̂@3&−16&7@3&8&−5)| = 𝑖 ̂ (–16(−5) – 8(7)) – 𝑗 ̂ (3(-5) – 3(7)) + 𝑘 ̂(3(8) – 3(–16)) = 𝑖 ̂ (80 – 56) – 𝑗 ̂ (–15 – 21) + 𝑘 ̂(24 + 48) = 24𝒊 ̂ + 36𝒋 ̂ + 72𝒌 ̂ Thus, 𝒃 ⃗ = 24𝒊 ̂ + 36𝒋 ̂ + 72𝒌 ̂ Now, Putting value of 𝑎 ⃗ & 𝑏 ⃗ in formula 𝑟 ⃗ = 𝑎 ⃗ + 𝜆𝑏 ⃗ ∴ 𝑟 ⃗ = (1𝒊 ̂ + 2𝒋 ̂ – 4𝒌 ̂) + 𝜆 (24𝒊 ̂ + 36𝒋 ̂ + 72𝒌 ̂) = (𝑖 ̂ + 2𝑗 ̂ – 4𝑘 ̂) + 𝜆12 (2𝑖 ̂ + 3𝑗 ̂ + 6𝑘 ̂) = (𝑖 ̂ + 2𝑗 ̂ – 4𝑘 ̂) + 𝜆 (2𝑖 ̂ + 3𝑗 ̂ + 6𝑘 ̂) Therefore, the equation of the line is (𝒊 ̂ + 2𝒋 ̂ – 4𝒌 ̂) + 𝜆 (2𝒊 ̂ + 3𝒋 ̂ + 6𝒌 ̂). Misc 5 (Method 2) Find the vector equation of the line passing through the point (1, 2, –4) and perpendicular to the two lines: (𝑥 − 8)/3 = (𝑦 + 19)/(−16) = (𝑧 − 10)/7 and (𝑥 − 15)/3 = (𝑦 − 29)/8 = (𝑧 − 5)/(−5) The vector equation of a line passing through a point with position vector 𝑎 ⃗ and parallel to a vector 𝑏 ⃗ is 𝒓 ⃗ = 𝒂 ⃗ + 𝜆𝒃 ⃗ The line passes through (1, 2, −4) So, 𝒂 ⃗ = 1𝒊 ̂ + 2𝒋 ̂ − 4𝒌 ̂ Let 𝒃 ⃗ = x𝒊 ̂ + y𝒋 ̂ + z𝒌 ̂ Two lines with direction ratios 𝑎1 , 𝑏1 , 𝑐1 & 𝑎2 , 𝑏2 , 𝑐2 are perpendicular if 𝒂𝟏 𝒂𝟐 + 𝒃𝟏𝒃𝟐 + 𝒄𝟏 𝒄𝟐 = 0 Given, line 𝑏 ⃗ is perpendicular to (𝑥 − 8)/3 = (𝑦 + 19)/16 = (𝑧 − 10)/7 and (𝑥 − 15)/3 = (𝑦 − 29)/8 = (𝑧 − 5)/( − 5) So, 3x − 16y + 7z = 0 and 3x + 8y − 5z = 0 𝑥/(80 − 56 ) = 𝑦/(21 − ( −15) ) = 𝑧/(24 − ( −48) ) 𝑥/(24 ) = 𝑦/36 = 𝑧/72 𝑥/2 = 𝑦/3 = 𝑧/6 = k Hence, x = 2k , y = 3k , & z = 6k Thus, 𝒃 ⃗ = x𝒊 ̂ + y𝒋 ̂ + z𝒌 ̂ = 2k𝒊 ̂ + 3k𝒋 ̂ + 6k𝒌 ̂ Now, Putting value of 𝑎 ⃗ & 𝑏 ⃗ in formula 𝑟 ⃗ = 𝑎 ⃗ + 𝜆𝑏 ⃗ ∴ 𝑟 ⃗ = (𝑖 ̂ + 2𝑗 ̂ − 4𝑘 ̂) + 𝜆 (2k𝑖 ̂ + 3k𝑗 ̂ + 6k𝑘 ̂) = (𝑖 ̂ + 2𝑗 ̂ − 4𝑘 ̂) + 𝜆k (2𝑖 ̂ + 3𝑗 ̂ + 6𝑘 ̂) = (𝑖 ̂ + 2𝑗 ̂ − 4𝑘 ̂) + 𝜆 (2𝑖 ̂ + 3𝑗 ̂ + 6𝑘 ̂) Therefore, the equation of the line is (𝒊 ̂ + 2𝒋 ̂ − 4𝒌 ̂) + 𝜆(2𝒊 ̂ + 3𝒋 ̂ + 6𝒌 ̂)