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If u and v are unit vectors and θ is the acute angle between them, then 2u × 3v is a unit vector for
- a)Exactly two values of θ
- b)More than two values of θ
- c)No value of θ
- d)Exactly one value of θ
Correct answer is option 'D'. Can you explain this answer?
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If u and v are unit vectors and θis the acuteangle between them,...
If u and v are unit vectors, it means that their magnitudes are equal to 1. Thus, ||u|| = ||v|| = 1.
Given that ||u + v|| = 2, we want to prove that ||u - v|| = 2.
Using the triangle inequality, we have:
||u + v|| ≤ ||u|| + ||v||
Since ||u|| = ||v|| = 1, the inequality becomes:
||u + v|| ≤ 1 + 1
||u + v|| ≤ 2
Since we are given that ||u + v|| = 2, the inequality becomes:
2 ≤ 2
This inequality is true, so the triangle inequality holds.
Now, let's consider ||u - v||:
||u - v|| = ||u + (-v)||
Since -v is the additive inverse of v, its magnitude is the same as v's magnitude, so ||-v|| = ||v|| = 1.
Using the triangle inequality again, we have:
||u + (-v)|| ≤ ||u|| + ||-v||
Since ||u|| = 1 and ||-v|| = 1, the inequality becomes:
||u + (-v)|| ≤ 1 + 1
||u + (-v)|| ≤ 2
However, since we are given that ||u + v|| = 2, we can replace u + (-v) with u + v in the inequality:
||u + v|| ≤ 2
Therefore, ||u - v|| ≤ 2.
But we also know that ||u - v|| ≥ 0, since magnitudes are always non-negative.
Since ||u - v|| ≤ 2 and ||u - v|| ≥ 0, the only possible value for ||u - v|| is 2.
Therefore, ||u - v|| = 2.
In conclusion, if u and v are unit vectors and ||u + v|| = 2, then ||u - v|| = 2.
Given that ||u + v|| = 2, we want to prove that ||u - v|| = 2.
Using the triangle inequality, we have:
||u + v|| ≤ ||u|| + ||v||
Since ||u|| = ||v|| = 1, the inequality becomes:
||u + v|| ≤ 1 + 1
||u + v|| ≤ 2
Since we are given that ||u + v|| = 2, the inequality becomes:
2 ≤ 2
This inequality is true, so the triangle inequality holds.
Now, let's consider ||u - v||:
||u - v|| = ||u + (-v)||
Since -v is the additive inverse of v, its magnitude is the same as v's magnitude, so ||-v|| = ||v|| = 1.
Using the triangle inequality again, we have:
||u + (-v)|| ≤ ||u|| + ||-v||
Since ||u|| = 1 and ||-v|| = 1, the inequality becomes:
||u + (-v)|| ≤ 1 + 1
||u + (-v)|| ≤ 2
However, since we are given that ||u + v|| = 2, we can replace u + (-v) with u + v in the inequality:
||u + v|| ≤ 2
Therefore, ||u - v|| ≤ 2.
But we also know that ||u - v|| ≥ 0, since magnitudes are always non-negative.
Since ||u - v|| ≤ 2 and ||u - v|| ≥ 0, the only possible value for ||u - v|| is 2.
Therefore, ||u - v|| = 2.
In conclusion, if u and v are unit vectors and ||u + v|| = 2, then ||u - v|| = 2.
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If u and v are unit vectors and θis the acuteangle between them, then 2u × 3v is a unit vector fora)Exactly two values ofθb)More than two values ofθc)No value of θd)Exactly one value of θCorrect answer is option 'D'. Can you explain this answer?
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If u and v are unit vectors and θis the acuteangle between them, then 2u × 3v is a unit vector fora)Exactly two values ofθb)More than two values ofθc)No value of θd)Exactly one value of θCorrect 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 If u and v are unit vectors and θis the acuteangle between them, then 2u × 3v is a unit vector fora)Exactly two values ofθb)More than two values ofθc)No value of θd)Exactly one value of θCorrect 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 If u and v are unit vectors and θis the acuteangle between them, then 2u × 3v is a unit vector fora)Exactly two values ofθb)More than two values ofθc)No value of θd)Exactly one value of θCorrect answer is option 'D'. Can you explain this answer?.
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