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The value of b such that the scalar product of the vector î+ĵ+k̂ with the unit vector parallel to the sum of the vectors 2î + 4ĵ - 5k̂ and bî+ 2ĵ + 3k̂ is one is
- a)-2
- b)-1
- c)0
- d)1
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
The value of b such that the scalar product of the vector î+j...
Parallel vector =(2+b)i+6j−2k
Unit vector = (2+b)i+6j−2k / (b2+4b+44)1/2
According to the question,
1 = (2+b)+6−2/b2+4b+44
⇒b2+4b+44=b2+12b+36
⇒8b=8⇒b=1
Most Upvoted Answer
The value of b such that the scalar product of the vector î+j...
Given vector i j k and two vectors 2i 4j - 5k and bi 2j 3k.
Let the unit vector parallel to the sum of the vectors 2i 4j - 5k and bi 2j 3k be represented as a unit vector U.
So, U = (2i + bi) / sqrt(4 + b^2) + (4j + 2j) / sqrt(16 + 4) - 5k / sqrt(25)
= (2i + bi) / sqrt(b^2 + 4) + 6j / sqrt(20) - k / 5
As U is a unit vector, |U| = 1
=> |U|^2 = 1
=> (2i + bi)^2 / (b^2 + 4) + 36 / 20 + 1 / 25 = 1
=> 4 + 4bi + b^2 / (b^2 + 4) + 9 / 5 + 1 / 25 = 1
=> b^2 + 4bi + 4 / (b^2 + 4) + 9 / 5 + 1 / 25 = 1
=> (b^2 + 4)^2 + 20bi(b^2 + 4) + 100 / 25(b^2 + 4) + 225(b^2 + 4) / 25 + 1 = 25 / 25
=> (b^2 + 4)^2 + 20bi(b^2 + 4) + 325 = 0
Now, as the scalar product of i j k with U is 1
=> (2i + bi) / sqrt(b^2 + 4) + 6j / sqrt(20) - 5k / sqrt(25) . i j k = 1
=> 10 / sqrt(20) = 1
=> sqrt(20) = 10
=> 20 = 100
As this is not possible, we can conclude that the scalar product cannot be 1 for any value of b.
Therefore, the given question has no solution.
Hence, the correct option is (D) None of these.
Let the unit vector parallel to the sum of the vectors 2i 4j - 5k and bi 2j 3k be represented as a unit vector U.
So, U = (2i + bi) / sqrt(4 + b^2) + (4j + 2j) / sqrt(16 + 4) - 5k / sqrt(25)
= (2i + bi) / sqrt(b^2 + 4) + 6j / sqrt(20) - k / 5
As U is a unit vector, |U| = 1
=> |U|^2 = 1
=> (2i + bi)^2 / (b^2 + 4) + 36 / 20 + 1 / 25 = 1
=> 4 + 4bi + b^2 / (b^2 + 4) + 9 / 5 + 1 / 25 = 1
=> b^2 + 4bi + 4 / (b^2 + 4) + 9 / 5 + 1 / 25 = 1
=> (b^2 + 4)^2 + 20bi(b^2 + 4) + 100 / 25(b^2 + 4) + 225(b^2 + 4) / 25 + 1 = 25 / 25
=> (b^2 + 4)^2 + 20bi(b^2 + 4) + 325 = 0
Now, as the scalar product of i j k with U is 1
=> (2i + bi) / sqrt(b^2 + 4) + 6j / sqrt(20) - 5k / sqrt(25) . i j k = 1
=> 10 / sqrt(20) = 1
=> sqrt(20) = 10
=> 20 = 100
As this is not possible, we can conclude that the scalar product cannot be 1 for any value of b.
Therefore, the given question has no solution.
Hence, the correct option is (D) None of these.
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The value of b such that the scalar product of the vector î+ĵ+k̂ with the unit vector parallel to the sum of the vectors 2î + 4ĵ - 5k̂ and bî+ 2ĵ + 3k̂ is one isa)-2b)-1c)0d)1Correct answer is option 'D'. Can you explain this answer?
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The value of b such that the scalar product of the vector î+ĵ+k̂ with the unit vector parallel to the sum of the vectors 2î + 4ĵ - 5k̂ and bî+ 2ĵ + 3k̂ is one isa)-2b)-1c)0d)1Correct answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The value of b such that the scalar product of the vector î+ĵ+k̂ with the unit vector parallel to the sum of the vectors 2î + 4ĵ - 5k̂ and bî+ 2ĵ + 3k̂ is one isa)-2b)-1c)0d)1Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The value of b such that the scalar product of the vector î+ĵ+k̂ with the unit vector parallel to the sum of the vectors 2î + 4ĵ - 5k̂ and bî+ 2ĵ + 3k̂ is one isa)-2b)-1c)0d)1Correct answer is option 'D'. Can you explain this answer?.
The value of b such that the scalar product of the vector î+ĵ+k̂ with the unit vector parallel to the sum of the vectors 2î + 4ĵ - 5k̂ and bî+ 2ĵ + 3k̂ is one isa)-2b)-1c)0d)1Correct answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The value of b such that the scalar product of the vector î+ĵ+k̂ with the unit vector parallel to the sum of the vectors 2î + 4ĵ - 5k̂ and bî+ 2ĵ + 3k̂ is one isa)-2b)-1c)0d)1Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The value of b such that the scalar product of the vector î+ĵ+k̂ with the unit vector parallel to the sum of the vectors 2î + 4ĵ - 5k̂ and bî+ 2ĵ + 3k̂ is one isa)-2b)-1c)0d)1Correct answer is option 'D'. Can you explain this answer?.
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