Check sibling questions

Assertion (A): The acute angle between the line

r = i Μ‚+j Μ‚+2k Μ‚+λ(i Μ‚-j Μ‚ ) and the x-axis is π/4

Reason(R): The acute angle πœƒ between the lines

 r = x 1 i Μ‚+y 1 j Μ‚+z 1 k Μ‚+λ(a 1 i Μ‚+b 1 j Μ‚+c 1 k Μ‚ )  and

 r  = x 1 i Μ‚+y 1 j Μ‚+z 1 k Μ‚+μ(a 1 i Μ‚+b 1 j Μ‚+c 1 k Μ‚ )is given by

π‘π‘œπ‘ πœƒ = |a 1 a 2 + b 1 b 2 + c 1 c 2 |/(√a 1 2 + b 1 2 + c 1 2 √a 2 2 + b2 2 . + c 2 2

 

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Transcript

Question 20 Assertion (A): The acute angle between the line π‘Ÿ βƒ— = 𝑖 Μ‚+𝑗 Μ‚+2π‘˜ Μ‚+πœ†(𝑖 Μ‚βˆ’π‘— Μ‚ ) and the x-axis is πœ‹/4 Reason(R): The acute angle πœƒ between the lines π‘Ÿ βƒ— = π‘₯_1 𝑖 Μ‚+𝑦_1 𝑗 Μ‚+𝑧_1 π‘˜ Μ‚+πœ†(π‘Ž_1 𝑖 Μ‚+𝑏_1 𝑗 Μ‚+𝑐_1 π‘˜ Μ‚ ) and π‘Ÿ βƒ— = π‘₯_2 𝑖 Μ‚+𝑦_2 𝑗 Μ‚+𝑧_2 π‘˜ Μ‚+πœ‡(π‘Ž_2 𝑖 Μ‚+𝑏_2 𝑗 Μ‚+𝑐_2 π‘˜ Μ‚ )is given by π‘π‘œπ‘ πœƒ = |π‘Ž_1 π‘Ž_2 + 𝑏_1 𝑏_2 + γ€– 𝑐〗_1 𝑐_2 |/(√(γ€–π‘Ž1γ€—^2 + 〖𝑏1γ€—^2 + 〖𝑐1γ€—^2 ) √(γ€–π‘Ž2γ€—^2 + 〖𝑏2γ€—^2. + 〖𝑐2γ€—^2 )) Checking Assertion Assertion (A): The acute angle between the line π‘Ÿ βƒ— = 𝑖 Μ‚+𝑗 Μ‚+2π‘˜ Μ‚+πœ†(𝑖 Μ‚βˆ’π‘— Μ‚ ) and the x-axis is πœ‹/4 Equation of x-axis Let’s consider two points on x-axis – (a, 0, 0), and (0, 0, 0) Vector equation of a line passing though two points with position vectors π‘Ž βƒ— and 𝑏 βƒ— is 𝒓 βƒ— = (𝒂 ) βƒ— + πœ† (𝒃 βƒ— βˆ’ 𝒂 βƒ—) Here, (a, 0, 0) 𝒂 βƒ— = a𝑖 Μ‚ + 0𝑗 Μ‚ + 0π‘˜ Μ‚ (0, 0, 0) 𝒃 βƒ— = 0𝑖 Μ‚ + 0𝑗 Μ‚ + 0π‘˜ Μ‚ Thus, equation of line is 𝒓 βƒ— = (a𝑖 Μ‚ + 0𝑗 Μ‚ + 0π‘˜ Μ‚) + πœ† [(0𝑖 Μ‚+0𝑗+0π‘˜ Μ‚ ) βˆ’ (π‘Žπ‘– Μ‚ +0𝑗 Μ‚ + 20)] = a𝑖 Μ‚ + πœ† [βˆ’π‘Žπ‘– Μ‚ ] = (a βˆ’ πœ†a)π’Š Μ‚ Since (a βˆ’ πœ†a) is a constant, let (a βˆ’ πœ†a) = k = kπ’Š Μ‚ Now, we need to find angle between the line π‘Ÿ βƒ— = 𝑖 Μ‚+𝑗 Μ‚+2π‘˜ Μ‚+πœ†(𝑖 Μ‚βˆ’π‘— Μ‚ ) and the x-axis i.e. Angle between 𝒓 βƒ— = π’Š Μ‚+𝒋 Μ‚+πŸπ’Œ Μ‚+𝝀(π’Š Μ‚βˆ’π’‹ Μ‚ ) and 𝒓 βƒ— = kπ’Š Μ‚ Using formula from Reasoning π‘π‘œπ‘  πœƒ = |𝒂_𝟏 𝒂_𝟐 + 𝒃_𝟏 𝒃_𝟐 + γ€– 𝒄〗_𝟏 𝒄_𝟐 |/(√(γ€–π’‚πŸγ€—^𝟐 + γ€–π’ƒπŸγ€—^𝟐 + γ€–π’„πŸγ€—^𝟐 ) √(γ€–π’‚πŸγ€—^𝟐 + γ€–π’ƒπŸγ€—^𝟐. + γ€–π’„πŸγ€—^𝟐 )) Thus, equation of line is 𝒓 βƒ— = (a𝑖 Μ‚ + 0𝑗 Μ‚ + 0π‘˜ Μ‚) + πœ† [(0𝑖 Μ‚+0𝑗+0π‘˜ Μ‚ ) βˆ’ (π‘Žπ‘– Μ‚ +0𝑗 Μ‚ + 20)] = a𝑖 Μ‚ + πœ† [βˆ’π‘Žπ‘– Μ‚ ] = (a βˆ’ πœ†a)π’Š Μ‚ Since (a βˆ’ πœ†a) is a constant, let (a βˆ’ πœ†a) = k = kπ’Š Μ‚ Now, we need to find angle between the line π‘Ÿ βƒ— = 𝑖 Μ‚+𝑗 Μ‚+2π‘˜ Μ‚+πœ†(𝑖 Μ‚βˆ’π‘— Μ‚ ) and the x-axis i.e. Angle between 𝒓 βƒ— = π’Š Μ‚+𝒋 Μ‚+πŸπ’Œ Μ‚+𝝀(π’Š Μ‚βˆ’π’‹ Μ‚ ) and 𝒓 βƒ— = kπ’Š Μ‚ Using formula from Reasoning π‘π‘œπ‘  πœƒ = |𝒂_𝟏 𝒂_𝟐 + 𝒃_𝟏 𝒃_𝟐 + γ€– 𝒄〗_𝟏 𝒄_𝟐 |/(√(γ€–π’‚πŸγ€—^𝟐 + γ€–π’ƒπŸγ€—^𝟐 + γ€–π’„πŸγ€—^𝟐 ) √(γ€–π’‚πŸγ€—^𝟐 + γ€–π’ƒπŸγ€—^𝟐. + γ€–π’„πŸγ€—^𝟐 )) 𝒓 βƒ— = π’Š Μ‚+𝒋 Μ‚+πŸπ’Œ Μ‚+𝝀(π’Š Μ‚βˆ’π’‹ Μ‚ ) Comparing with π‘Ÿ βƒ— = π‘₯_1 𝑖 Μ‚+𝑦_1 𝑗 Μ‚+𝑧_1 π‘˜ Μ‚+πœ†(π‘Ž_1 𝑖 Μ‚+𝑏_1 𝑗 Μ‚+𝑐_1 π‘˜ Μ‚ ) 𝒂1 = 1, b1 = βˆ’1, c1 = 0 𝒓 βƒ—" = k" π’Š Μ‚ Comparing with π‘Ÿ βƒ— = π‘₯_2 𝑖 Μ‚+𝑦_2 𝑗 Μ‚+𝑧_2 π‘˜ Μ‚+πœ†(π‘Ž_2 𝑖 Μ‚+𝑏_2 𝑗 Μ‚+𝑐_2 π‘˜ Μ‚ ) 𝒂2 = 1, 𝒃2 = 0, 𝒄2 = 0 Now, cos ΞΈ = |(π‘Ž_1 π‘Ž_2 + 𝑏_1 𝑏_2 +γ€– 𝑐〗_1 𝑐_2)/(√(γ€–π‘Ž_1γ€—^2 + 〖𝑏_1γ€—^2+ 〖𝑐_1γ€—^2 ) √(γ€–π‘Ž_2γ€—^2 +γ€–γ€– 𝑏〗_2γ€—^2+ 〖𝑐_2γ€—^2 ))| = |((1 Γ— 1) + (βˆ’1 Γ— 0) + (0 Γ— 0))/(√(1^2 +(βˆ’1)^2 + 0^2 ) Γ— √(1^2 + 0^2 + 0^2 ))| = |1/(√(1 + 1) Γ— √1)| = |1/√2| = 𝟏/√𝟐 So, cos ΞΈ = 1/√2 ∴ ΞΈ = πœ‹/πŸ’ Therefore, the angle between the given pair of line is πœ‹/πŸ’ So, Assertion is true Checking Reason Reason(R): The acute angle πœƒ between the lines π‘Ÿ βƒ— = π‘₯_1 𝑖 Μ‚+𝑦_1 𝑗 Μ‚+𝑧_1 π‘˜ Μ‚+πœ†(π‘Ž_1 𝑖 Μ‚+𝑏_1 𝑗 Μ‚+𝑐_1 π‘˜ Μ‚ ) and π‘Ÿ βƒ— = π‘₯_2 𝑖 Μ‚+𝑦_2 𝑗 Μ‚+𝑧_2 π‘˜ Μ‚+πœ‡(π‘Ž_2 𝑖 Μ‚+𝑏_2 𝑗 Μ‚+𝑐_2 π‘˜ Μ‚ )is given by π‘π‘œπ‘ πœƒ = |π‘Ž_1 π‘Ž_2 + 𝑏_1 𝑏_2 + γ€– 𝑐〗_1 𝑐_2 |/(√(γ€–π‘Ž1γ€—^2 + 〖𝑏1γ€—^2 + 〖𝑐1γ€—^2 ) √(γ€–π‘Ž2γ€—^2 + 〖𝑏2γ€—^2. + 〖𝑐2γ€—^2 )) This is a formula and it is a correct formula Thus, Reasoning is true Is Reason a Correct explanation for Assertion? Sine we used formula mentioned in Reasoning to find Assertion Therefore, Reasoning is a correct explanation for Assertion So, Assertion is true Reasoning is true And, Reasoning is a correct explanation for Assertion So, the correct answer is (a)

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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.