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What is the value of b such that the scalar product of the vector i + j + k with the unit vector parallel to the sum of the vectors 2i + 4j – 5k and bi + 2j + 3k is unity?
- a)–2
- b)–1
- c)0
- d)1
Correct answer is option 'D'. Can you explain this answer?
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To find the value of b, we need to calculate the scalar product of the vector i + j + k with the unit vector parallel to the sum of the vectors 2i + 4j + 5k and bi + 2j + 3k and set it equal to unity.
Given vectors:
Vector a = i + j + k
Vector b = 2i + 4j + 5k
Vector c = bi + 2j + 3k
We need to find the value of b such that the scalar product of a and the unit vector parallel to the sum of b and c is equal to 1.
Let's break down the problem step by step:
Step 1: Find the sum of vectors b and c.
Sum of vectors b and c = (2i + 4j + 5k) + (bi + 2j + 3k) = (2 + b)i + (4 + 2)j + (5 + 3)k = (2 + b)i + 6j + 8k
Step 2: Find the unit vector parallel to the sum of vectors b and c.
To find the unit vector, we divide the sum of vectors by its magnitude.
Magnitude of the sum of vectors = √((2 + b)^2 + 6^2 + 8^2) = √(4 + 4b + b^2 + 36 + 64) = √(b^2 + 4b + 104)
Unit vector parallel to the sum of vectors = ((2 + b)i + 6j + 8k) / √(b^2 + 4b + 104)
Step 3: Calculate the scalar product between vector a and the unit vector.
Scalar product = a · ((2 + b)i + 6j + 8k)
= (1)(2 + b) + (1)(6) + (1)(8)
= 2 + b + 6 + 8
= b + 16
Step 4: Set the scalar product equal to 1 and solve for b.
b + 16 = 1
b = 1 - 16
b = -15
Therefore, the value of b is -15, which is not among the given options. It seems there might be an error in the answer choices provided for this question.
Given vectors:
Vector a = i + j + k
Vector b = 2i + 4j + 5k
Vector c = bi + 2j + 3k
We need to find the value of b such that the scalar product of a and the unit vector parallel to the sum of b and c is equal to 1.
Let's break down the problem step by step:
Step 1: Find the sum of vectors b and c.
Sum of vectors b and c = (2i + 4j + 5k) + (bi + 2j + 3k) = (2 + b)i + (4 + 2)j + (5 + 3)k = (2 + b)i + 6j + 8k
Step 2: Find the unit vector parallel to the sum of vectors b and c.
To find the unit vector, we divide the sum of vectors by its magnitude.
Magnitude of the sum of vectors = √((2 + b)^2 + 6^2 + 8^2) = √(4 + 4b + b^2 + 36 + 64) = √(b^2 + 4b + 104)
Unit vector parallel to the sum of vectors = ((2 + b)i + 6j + 8k) / √(b^2 + 4b + 104)
Step 3: Calculate the scalar product between vector a and the unit vector.
Scalar product = a · ((2 + b)i + 6j + 8k)
= (1)(2 + b) + (1)(6) + (1)(8)
= 2 + b + 6 + 8
= b + 16
Step 4: Set the scalar product equal to 1 and solve for b.
b + 16 = 1
b = 1 - 16
b = -15
Therefore, the value of b is -15, which is not among the given options. It seems there might be an error in the answer choices provided for this question.
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What is the value of b such that the scalar product of the vector i + j + k with the unit vector parallel to the sum of the vectors 2i + 4j 5k and bi + 2j + 3k is unity?a)2b)1c)0d)1Correct answer is option 'D'. Can you explain this answer?
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What is the value of b such that the scalar product of the vector i + j + k with the unit vector parallel to the sum of the vectors 2i + 4j 5k and bi + 2j + 3k is unity?a)2b)1c)0d)1Correct answer is option 'D'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about What is the value of b such that the scalar product of the vector i + j + k with the unit vector parallel to the sum of the vectors 2i + 4j 5k and bi + 2j + 3k is unity?a)2b)1c)0d)1Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What is the value of b such that the scalar product of the vector i + j + k with the unit vector parallel to the sum of the vectors 2i + 4j 5k and bi + 2j + 3k is unity?a)2b)1c)0d)1Correct answer is option 'D'. Can you explain this answer?.
What is the value of b such that the scalar product of the vector i + j + k with the unit vector parallel to the sum of the vectors 2i + 4j 5k and bi + 2j + 3k is unity?a)2b)1c)0d)1Correct answer is option 'D'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about What is the value of b such that the scalar product of the vector i + j + k with the unit vector parallel to the sum of the vectors 2i + 4j 5k and bi + 2j + 3k is unity?a)2b)1c)0d)1Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What is the value of b such that the scalar product of the vector i + j + k with the unit vector parallel to the sum of the vectors 2i + 4j 5k and bi + 2j + 3k is unity?a)2b)1c)0d)1Correct answer is option 'D'. Can you explain this answer?.
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