Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class
Miscellaneous
Misc 2 Important
Misc 3 Important
Misc 4 Important
Misc 5 Important
Question 1 Important Deleted for CBSE Board 2024 Exams
Question 2 Deleted for CBSE Board 2024 Exams
Question 3 Important Deleted for CBSE Board 2024 Exams
Question 4 Deleted for CBSE Board 2024 Exams
Question 5 Important Deleted for CBSE Board 2024 Exams
Question 6 Deleted for CBSE Board 2024 Exams
Question 7 Important Deleted for CBSE Board 2024 Exams
Question 8 Important Deleted for CBSE Board 2024 Exams
Question 9 Important Deleted for CBSE Board 2024 Exams
Question 10 Important Deleted for CBSE Board 2024 Exams
Question 11 Important Deleted for CBSE Board 2024 Exams
Question 12 Deleted for CBSE Board 2024 Exams
Question 13 Important Deleted for CBSE Board 2024 Exams
Question 14 Important Deleted for CBSE Board 2024 Exams
Question 15 Deleted for CBSE Board 2024 Exams You are here
Question 16 Important Deleted for CBSE Board 2024 Exams
Question 17 (MCQ) Important Deleted for CBSE Board 2024 Exams
Question 18 (MCQ) Important Deleted for CBSE Board 2024 Exams
Miscellaneous
Last updated at May 29, 2023 by Teachoo
Question 15 (Method 1) Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes 𝑟 ⃗ . (𝑖 ̂ − 𝑗 ̂ + 2 𝑘 ̂) = 5 and 𝑟 ⃗ . (3𝑖 ̂ + 𝑗 ̂ + 𝑘 ̂) = 6 . The vector equation of a line passing through a point with position vector 𝑎 ⃗ and parallel to a vector 𝑏 ⃗ is 𝒓 ⃗ = 𝒂 ⃗ + 𝜆𝒃 ⃗ Given, the line passes through (1, 2, 3) So, 𝑎 ⃗ = 1𝑖 ̂ + 2𝑗 ̂ + 3𝑘 ̂ Given, line is parallel to both planes ∴ Line is perpendicular to normal of both planes. i.e. 𝑏 ⃗ is perpendicular to normal of both planes. The vector equation of a line passing through a point with position vector 𝑎 ⃗ and parallel to a vector 𝑏 ⃗ is 𝒓 ⃗ = 𝒂 ⃗ + 𝜆𝒃 ⃗ Given, the line passes through (1, 2, 3) So, 𝑎 ⃗ = 1𝑖 ̂ + 2𝑗 ̂ + 3𝑘 ̂ Given, line is parallel to both planes ∴ Line is perpendicular to normal of both planes. i.e. 𝑏 ⃗ is perpendicular to normal of both planes. We know that 𝑎 ⃗ × 𝑏 ⃗ is perpendicular to both 𝑎 ⃗ & 𝑏 ⃗ So, 𝑏 ⃗ is cross product of normal of planes 𝑟 ⃗ . (𝑖 ̂ − 𝑗 ̂ + 2 𝑘 ̂) = 5 and 𝑟 ⃗ . (3𝑖 ̂ + 𝑗 ̂ + 𝑘 ̂) = 6 Required normal = |■8(𝑖 ̂&𝑗 ̂&𝑘 ̂@1&−1&2@3&1&1)| = 𝑖 ̂ (–1(1) – 1(2)) – 𝑗 ̂ (1(1) – 3(2)) + 𝑘 ̂(1(1) – 3(–1)) = 𝑖 ̂ (–1 – 2) – 𝑗 ̂ (1 – 6) + 𝑘 ̂(1 + 3) = –3𝑖 ̂ + 5𝑗 ̂ + 4𝑘 ̂ Thus, 𝑏 ⃗ = −3𝑖 ̂ + 5𝑗 ̂ + 4𝑘 ̂ Now, Putting value of 𝑎 ⃗ & 𝑏 ⃗ in formula 𝑟 ⃗ = 𝑎 ⃗ + 𝜆𝑏 ⃗ = (𝒊 ̂ + 2𝒋 ̂ + 3𝒌 ̂) + 𝜆 (−3𝒊 ̂ + 5𝒋 ̂ + 4𝒌 ̂) Now, Putting value of 𝑎 ⃗ & 𝑏 ⃗ in formula 𝑟 ⃗ = 𝑎 ⃗ + 𝜆𝑏 ⃗ = (𝒊 ̂ + 2𝒋 ̂ + 3𝒌 ̂) + 𝜆 (−3𝒊 ̂ + 5𝒋 ̂ + 4𝒌 ̂) Question 15 (Method 2) Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes 𝑟 ⃗ . (𝑖 ̂ − 𝑗 ̂ + 2 𝑘 ̂) = 5 and 𝑟 ⃗ . (3𝑖 ̂ + 𝑗 ̂ + 𝑘 ̂) = 6 . The vector equation of a line passing through a point with position vector 𝑎 ⃗ and parallel to a vector 𝑏 ⃗ is 𝒓 ⃗ = 𝒂 ⃗ + 𝜆𝒃 ⃗ Given, the line passes through (1, 2, 3) So, 𝑎 ⃗ = 1𝑖 ̂ + 2𝑗 ̂ + 3𝑘 ̂ Let 𝑏 ⃗ = 𝑏_1 𝑖 ̂ + 𝑏_2 𝑗 ̂ + 𝑏_3 𝑘 ̂ A line parallel to a plane is perpendicular to the normal of the plane. And two lines 𝑝 ⃗ and 𝑞 ⃗ are perpendicular if 𝑝 ⃗.𝑞 ⃗ = 0 Given, the line is parallel to planes 𝒓 ⃗.(𝒊 ̂ − 𝒋 ̂ + 2𝒌 ̂) = 5 Comparing with 𝑟 ⃗. (𝑛1) ⃗ = d1, (𝑛1) ⃗ = 1𝑖 ̂ − 1𝑗 ̂ + 2𝑘 ̂ Since 𝑏 ⃗ is ⊥ to (𝑛1) ⃗, 𝑏 ⃗.(𝑛1) ⃗ = 0 (𝑏1𝑖 ̂ + 𝑏2 𝑗 ̂ + 𝑏3 𝑘 ̂).(1𝑖 ̂ − 1𝑗 ̂ + 2𝑘 ̂) = 0 (𝑏"1"× 1) + (𝑏"2"× −1) + (𝑏3 × 2) = 0 𝒃1 − 𝒃2 + 2𝒃3 = 0 𝒓 ⃗.(3𝒊 ̂ + 𝒋 ̂ + 𝒌 ̂) = 6 Comparing with 𝑟 ⃗. (𝑛2) ⃗ = d2, (𝑛2) ⃗ = 3𝑖 ̂ + 1𝑗 ̂ + 1𝑘 ̂ Since 𝑏 ⃗ is ⊥ to (𝑛2) ⃗, 𝑏 ⃗.(𝑛2) ⃗ = 0 (𝑏1 𝑖 ̂ + 𝑏2 𝑗 ̂ + 𝑏3 𝑘 ̂).(3𝑖 ̂ + 1𝑗 ̂ + 1𝑘 ̂) = 0 (𝑏1 × 3) + (𝑏2 × 1) + (𝑏3 × 1) = 0 3𝒃1 + 𝒃2 + 𝒃3 = 0 So, our equations are 𝑏1 − 𝑏2 + 2𝑏3 = 0 3𝑏1 + 𝑏2 + 𝑏3 = 0 Thus, 𝑏 ⃗ = 𝑏_1 𝑖 ̂ + 𝑏_2 𝑗 ̂ + 𝑏_3 𝑘 ̂ = −3k𝑖 ̂ + 5k𝑗 ̂ + 4k𝑘 ̂ Now, Putting value of 𝑎 ⃗ & 𝑏 ⃗ in formula 𝑟 ⃗ = 𝑎 ⃗ + 𝜆𝑏 ⃗ ∴ 𝑟 ⃗ = (1𝑖 ̂ + 2𝑗 ̂ + 3𝑘 ̂) + 𝜆 (−3k𝑖 ̂ + 5k𝑗 ̂ + 4k𝑘 ̂) = (𝑖 ̂ + 2𝑗 ̂ + 3𝑘 ̂) + 𝜆k (−3𝑖 ̂ + 5𝑗 ̂ + 4𝑘 ̂) = (𝑖 ̂ + 2𝑗 ̂ + 3𝑘 ̂) + 𝜆 (−3𝑖 ̂ + 5𝑗 ̂ + 4𝑘 ̂) Therefore, the equation of the line is (𝒊 ̂ + 2𝒋 ̂ + 3𝒌 ̂) + 𝜆 (−3𝒊 ̂ + 5𝒋 ̂ + 4𝒌 ̂).