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can anyone hlp me with thiss...a vector a has components 2p nd 1 with rspct 2 rectangular Cartesian coordinate system.this system is rotated through a certain angle about the origin in counter clockwise sense. if,with respect to new system ,a has components p+1 and 1.p=?
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can anyone hlp me with thiss...a vector a has components 2p nd 1 with ...
Introduction:
In this problem, we are given a vector a with components 2p and 1 in a rectangular Cartesian coordinate system. We need to find the value of p when the coordinate system is rotated about the origin in a counterclockwise direction.
Solution:
Step 1: Understanding the Rotation:
When a coordinate system is rotated about the origin, the x and y axes change their orientation. The new x' and y' axes are obtained by rotating the original x and y axes by a certain angle in the counterclockwise direction.
Step 2: Expressing Vector a in the New Coordinate System:
To express vector a in the new coordinate system, we need to determine its components along the new x' and y' axes.
Let's denote the angle of rotation as θ.
Using the rules of trigonometry, we can express the new components of vector a as follows:
ax' = axcos(θ) - aysin(θ)
ay' = axsin(θ) + aycos(θ)
Substituting the given components of vector a (2p and 1), we have:
p = (2p)cos(θ) - 1sin(θ)
1 = (2p)sin(θ) + 1cos(θ)
Step 3: Solving for p:
We have obtained two equations involving p and θ. To find the value of p, we need to solve these equations simultaneously.
p = 2pcos(θ) - sin(θ) --(1)
1 = 2psin(θ) + cos(θ) --(2)
From equation (2), we can express sin(θ) in terms of cos(θ):
sin(θ) = 1 - 2psin(θ)
Substituting this back into equation (1):
p = 2pcos(θ) - (1 - 2psin(θ))
Simplifying further:
p = 2pcos(θ) - 1 + 2p(1 - 2psin(θ))
p = 2p(2cos(θ) - 2sin(θ)) - 1 + 2p
p = 4p(cos(θ) - sin(θ)) - 1 + 2p
p = 4p(cos(θ) - sin(θ)) + (2p - 1)
Comparing the coefficients of p on both sides, we have:
1 = 4(cos(θ) - sin(θ))
cos(θ) - sin(θ) = 1/4
Step 4: Solving for θ:
To find the value of θ, we solve the equation cos(θ) - sin(θ) = 1/4.
Using trigonometric identities, we can express cos(θ) and sin(θ) in terms of tan(θ):
cos(θ)
In this problem, we are given a vector a with components 2p and 1 in a rectangular Cartesian coordinate system. We need to find the value of p when the coordinate system is rotated about the origin in a counterclockwise direction.
Solution:
Step 1: Understanding the Rotation:
When a coordinate system is rotated about the origin, the x and y axes change their orientation. The new x' and y' axes are obtained by rotating the original x and y axes by a certain angle in the counterclockwise direction.
Step 2: Expressing Vector a in the New Coordinate System:
To express vector a in the new coordinate system, we need to determine its components along the new x' and y' axes.
Let's denote the angle of rotation as θ.
Using the rules of trigonometry, we can express the new components of vector a as follows:
ax' = axcos(θ) - aysin(θ)
ay' = axsin(θ) + aycos(θ)
Substituting the given components of vector a (2p and 1), we have:
p = (2p)cos(θ) - 1sin(θ)
1 = (2p)sin(θ) + 1cos(θ)
Step 3: Solving for p:
We have obtained two equations involving p and θ. To find the value of p, we need to solve these equations simultaneously.
p = 2pcos(θ) - sin(θ) --(1)
1 = 2psin(θ) + cos(θ) --(2)
From equation (2), we can express sin(θ) in terms of cos(θ):
sin(θ) = 1 - 2psin(θ)
Substituting this back into equation (1):
p = 2pcos(θ) - (1 - 2psin(θ))
Simplifying further:
p = 2pcos(θ) - 1 + 2p(1 - 2psin(θ))
p = 2p(2cos(θ) - 2sin(θ)) - 1 + 2p
p = 4p(cos(θ) - sin(θ)) - 1 + 2p
p = 4p(cos(θ) - sin(θ)) + (2p - 1)
Comparing the coefficients of p on both sides, we have:
1 = 4(cos(θ) - sin(θ))
cos(θ) - sin(θ) = 1/4
Step 4: Solving for θ:
To find the value of θ, we solve the equation cos(θ) - sin(θ) = 1/4.
Using trigonometric identities, we can express cos(θ) and sin(θ) in terms of tan(θ):
cos(θ)
Community Answer
can anyone hlp me with thiss...a vector a has components 2p nd 1 with ...
Since the axes are rotated, the magnitude of the vector won't change.
so √{(p+1)^2 + 1} = √{(2p)^2 + 1}
solving this one may get,
p = 1 or -1/3
so √{(p+1)^2 + 1} = √{(2p)^2 + 1}
solving this one may get,
p = 1 or -1/3
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can anyone hlp me with thiss...a vector a has components 2p nd 1 with rspct 2 rectangular Cartesian coordinate system.this system is rotated through a certain angle about the origin in counter clockwise sense. if,with respect to new system ,a has components p+1 and 1.p=?
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can anyone hlp me with thiss...a vector a has components 2p nd 1 with rspct 2 rectangular Cartesian coordinate system.this system is rotated through a certain angle about the origin in counter clockwise sense. if,with respect to new system ,a has components p+1 and 1.p=? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about can anyone hlp me with thiss...a vector a has components 2p nd 1 with rspct 2 rectangular Cartesian coordinate system.this system is rotated through a certain angle about the origin in counter clockwise sense. if,with respect to new system ,a has components p+1 and 1.p=? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for can anyone hlp me with thiss...a vector a has components 2p nd 1 with rspct 2 rectangular Cartesian coordinate system.this system is rotated through a certain angle about the origin in counter clockwise sense. if,with respect to new system ,a has components p+1 and 1.p=?.
can anyone hlp me with thiss...a vector a has components 2p nd 1 with rspct 2 rectangular Cartesian coordinate system.this system is rotated through a certain angle about the origin in counter clockwise sense. if,with respect to new system ,a has components p+1 and 1.p=? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about can anyone hlp me with thiss...a vector a has components 2p nd 1 with rspct 2 rectangular Cartesian coordinate system.this system is rotated through a certain angle about the origin in counter clockwise sense. if,with respect to new system ,a has components p+1 and 1.p=? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for can anyone hlp me with thiss...a vector a has components 2p nd 1 with rspct 2 rectangular Cartesian coordinate system.this system is rotated through a certain angle about the origin in counter clockwise sense. if,with respect to new system ,a has components p+1 and 1.p=?.
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