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Two vectors of equal magnitudes P are inclined at some angle such that the difference in magnitude of resultant of either of the vectors is o. 732 times either of the magnitude of vectors. If the angle between them is increased by half of its initial value then the magnitude of difference of the vectors is √X P. Find the value of X.?
Verified Answer
Two vectors of equal magnitudes P are inclined at some angle such that...
R^2 = P^2 + Q^2 + 2PQ Cosθ
Here R = 1.732P and P=Q
So,
(1.732P)^2=P^2 + P^2 + 2P^2 Cosθ
3=2+2Cosθ
1=2Cosθ
Cosθ =1/2
θ = 60 degrees
"If the angle between them is increased to half of its initial" ==> I assume you meant "decreased"
If so,
R^2 = P^2 + P^2 + 2P^2 Cos(30 degrees)
R^2 = 2P^2 + 2P^2*(1.732)/2
R^2/P^2 = 2 + 1.732=3.732
R/P = sqrt(3.732)=1.932
R = 1.932P
So, now difference in the magnitude of resultant and magnitude of either of the vectors is 0.932 times either of the vectors
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Most Upvoted Answer
Two vectors of equal magnitudes P are inclined at some angle such that...
Given information:
- Two vectors have equal magnitudes P.
- The difference in magnitude of the resultant of either vector is 0.732 times either of the magnitudes of the vectors.
- The angle between the vectors is increased by half of its initial value.
- The magnitude of the difference of the vectors after the increase in angle is √X P.
To find:
The value of X.
Solution:
We can solve this problem by breaking it down into steps.
Step 1: Initial conditions
Let the initial angle between the vectors be θ.
The magnitude of the resultant of either vector is given as 0.732P.
Therefore, the magnitude of each vector is P.
Step 2: Finding the initial magnitudes
Using the Law of Cosines, we can find the initial magnitudes of the vectors.
The Law of Cosines states that for a triangle with sides a, b, and c, and angle C opposite side c, the following formula holds:
c^2 = a^2 + b^2 - 2abcosC
In our case, let the magnitudes of the vectors be a and b.
Using the Law of Cosines, we have:
(0.732P)^2 = P^2 + P^2 - 2P^2cosθ
0.536424P^2 = 2P^2 - 2P^2cosθ
0.464424P^2 = 2P^2cosθ
Step 3: Finding the initial angle
From the equation obtained in Step 2, we can solve for cosθ:
cosθ = 0.464424/2
cosθ = 0.232212
Taking the inverse cosine of both sides, we find:
θ = cos^(-1)(0.232212)
Step 4: Increasing the angle
The angle between the vectors is increased by half of its initial value, which means the new angle is:
θ_new = θ + 0.5θ
θ_new = 1.5θ
Step 5: Finding the new magnitudes
Using the Law of Cosines again, we can find the new magnitudes of the vectors.
(0.732P)^2 = P^2 + P^2 - 2P^2cosθ_new
0.536424P^2 = 2P^2 - 2P^2cos(1.5θ)
Step 6: Finding the magnitude difference
We are given that the magnitude of the difference of the vectors after the increase in angle is √X P.
Therefore, the magnitude of the difference is √(X)P.
From Step 5, we can see that the magnitude of the difference is 0.732P - (2P - 2Pcos(1.5θ)).
Simplifying further, we get:
0.732P - 2P + 2Pcos(1.5θ) = √(X)P
Step 7: Finding the value of X
From Step 6, we have the equation:
0.732P - 2P + 2Pcos(1.5θ) =
- Two vectors have equal magnitudes P.
- The difference in magnitude of the resultant of either vector is 0.732 times either of the magnitudes of the vectors.
- The angle between the vectors is increased by half of its initial value.
- The magnitude of the difference of the vectors after the increase in angle is √X P.
To find:
The value of X.
Solution:
We can solve this problem by breaking it down into steps.
Step 1: Initial conditions
Let the initial angle between the vectors be θ.
The magnitude of the resultant of either vector is given as 0.732P.
Therefore, the magnitude of each vector is P.
Step 2: Finding the initial magnitudes
Using the Law of Cosines, we can find the initial magnitudes of the vectors.
The Law of Cosines states that for a triangle with sides a, b, and c, and angle C opposite side c, the following formula holds:
c^2 = a^2 + b^2 - 2abcosC
In our case, let the magnitudes of the vectors be a and b.
Using the Law of Cosines, we have:
(0.732P)^2 = P^2 + P^2 - 2P^2cosθ
0.536424P^2 = 2P^2 - 2P^2cosθ
0.464424P^2 = 2P^2cosθ
Step 3: Finding the initial angle
From the equation obtained in Step 2, we can solve for cosθ:
cosθ = 0.464424/2
cosθ = 0.232212
Taking the inverse cosine of both sides, we find:
θ = cos^(-1)(0.232212)
Step 4: Increasing the angle
The angle between the vectors is increased by half of its initial value, which means the new angle is:
θ_new = θ + 0.5θ
θ_new = 1.5θ
Step 5: Finding the new magnitudes
Using the Law of Cosines again, we can find the new magnitudes of the vectors.
(0.732P)^2 = P^2 + P^2 - 2P^2cosθ_new
0.536424P^2 = 2P^2 - 2P^2cos(1.5θ)
Step 6: Finding the magnitude difference
We are given that the magnitude of the difference of the vectors after the increase in angle is √X P.
Therefore, the magnitude of the difference is √(X)P.
From Step 5, we can see that the magnitude of the difference is 0.732P - (2P - 2Pcos(1.5θ)).
Simplifying further, we get:
0.732P - 2P + 2Pcos(1.5θ) = √(X)P
Step 7: Finding the value of X
From Step 6, we have the equation:
0.732P - 2P + 2Pcos(1.5θ) =
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Two vectors of equal magnitudes P are inclined at some angle such that the difference in magnitude of resultant of either of the vectors is o. 732 times either of the magnitude of vectors. If the angle between them is increased by half of its initial value then the magnitude of difference of the vectors is √X P. Find the value of X.?
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Two vectors of equal magnitudes P are inclined at some angle such that the difference in magnitude of resultant of either of the vectors is o. 732 times either of the magnitude of vectors. If the angle between them is increased by half of its initial value then the magnitude of difference of the vectors is √X P. Find the value of X.? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Two vectors of equal magnitudes P are inclined at some angle such that the difference in magnitude of resultant of either of the vectors is o. 732 times either of the magnitude of vectors. If the angle between them is increased by half of its initial value then the magnitude of difference of the vectors is √X P. Find the value of X.? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two vectors of equal magnitudes P are inclined at some angle such that the difference in magnitude of resultant of either of the vectors is o. 732 times either of the magnitude of vectors. If the angle between them is increased by half of its initial value then the magnitude of difference of the vectors is √X P. Find the value of X.?.
Two vectors of equal magnitudes P are inclined at some angle such that the difference in magnitude of resultant of either of the vectors is o. 732 times either of the magnitude of vectors. If the angle between them is increased by half of its initial value then the magnitude of difference of the vectors is √X P. Find the value of X.? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Two vectors of equal magnitudes P are inclined at some angle such that the difference in magnitude of resultant of either of the vectors is o. 732 times either of the magnitude of vectors. If the angle between them is increased by half of its initial value then the magnitude of difference of the vectors is √X P. Find the value of X.? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two vectors of equal magnitudes P are inclined at some angle such that the difference in magnitude of resultant of either of the vectors is o. 732 times either of the magnitude of vectors. If the angle between them is increased by half of its initial value then the magnitude of difference of the vectors is √X P. Find the value of X.?.
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Two vectors of equal magnitudes P are inclined at some angle such that the difference in magnitude of resultant of either of the vectors is o. 732 times either of the magnitude of vectors. If the angle between them is increased by half of its initial value then the magnitude of difference of the vectors is √X P. Find the value of X.?, a detailed solution for Two vectors of equal magnitudes P are inclined at some angle such that the difference in magnitude of resultant of either of the vectors is o. 732 times either of the magnitude of vectors. If the angle between them is increased by half of its initial value then the magnitude of difference of the vectors is √X P. Find the value of X.? has been provided alongside types of Two vectors of equal magnitudes P are inclined at some angle such that the difference in magnitude of resultant of either of the vectors is o. 732 times either of the magnitude of vectors. If the angle between them is increased by half of its initial value then the magnitude of difference of the vectors is √X P. Find the value of X.? theory, EduRev gives you an
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