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20.The vector vec A=5hat i 6hat j is rotated through an angle 45^(@) about the z-axis in the anticlockwise direction.The resultant vector is H.C.U.-2016 (a) 11hat i 1hat j (b) ihat i 11hat j (c) (11)/(sqrt(2))hat i (1)/(sqrt(2))hat j?
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20.The vector vec A=5hat i 6hat j is rotated through an angle 45^(@) a...
Rotation of a Vector in 3D Space
To rotate a vector in 3D space, we can use a rotation matrix. In this case, the vector A is rotated about the z-axis in the counterclockwise direction by an angle of 45 degrees.
Rotation Matrix
The rotation matrix for a counterclockwise rotation about the z-axis can be written as:
R = [cosθ -sinθ 0]
[sinθ cosθ 0]
[ 0 0 1]
where θ is the angle of rotation.
Calculating the Resultant Vector
To calculate the resultant vector, we can multiply the rotation matrix R by the original vector A:
[R] x [A] = [cosθ -sinθ 0] x [5]
[6]
[0]
Multiplying the matrices, we get:
[R] x [A] = [(5cosθ - 6sinθ)
(6cosθ + 5sinθ)
0]
Substituting θ = 45 degrees, which is equal to π/4 radians, we can calculate the values:
[R] x [A] = [(5cos(π/4) - 6sin(π/4))
(6cos(π/4) + 5sin(π/4))
0]
Simplifying further, we get:
[R] x [A] = [(5√2/2 - 6√2/2)
(6√2/2 + 5√2/2)
0]
[R] x [A] = [(-√2 - 3√2)
(3√2 + √2)
0]
[R] x [A] = [-4√2
4√2
0]
Resultant Vector
The resultant vector is given by the components of the matrix:
Resultant vector = -4√2 hat i + 4√2 hat j + 0 hat k
Simplifying further, we get:
Resultant vector = -4√2 hat i + 4√2 hat j
The correct option is (c) (11√2)/(√2) hat i + (1√2)/(√2) hat j, which can be simplified as 11 hat i + hat j.
To rotate a vector in 3D space, we can use a rotation matrix. In this case, the vector A is rotated about the z-axis in the counterclockwise direction by an angle of 45 degrees.
Rotation Matrix
The rotation matrix for a counterclockwise rotation about the z-axis can be written as:
R = [cosθ -sinθ 0]
[sinθ cosθ 0]
[ 0 0 1]
where θ is the angle of rotation.
Calculating the Resultant Vector
To calculate the resultant vector, we can multiply the rotation matrix R by the original vector A:
[R] x [A] = [cosθ -sinθ 0] x [5]
[6]
[0]
Multiplying the matrices, we get:
[R] x [A] = [(5cosθ - 6sinθ)
(6cosθ + 5sinθ)
0]
Substituting θ = 45 degrees, which is equal to π/4 radians, we can calculate the values:
[R] x [A] = [(5cos(π/4) - 6sin(π/4))
(6cos(π/4) + 5sin(π/4))
0]
Simplifying further, we get:
[R] x [A] = [(5√2/2 - 6√2/2)
(6√2/2 + 5√2/2)
0]
[R] x [A] = [(-√2 - 3√2)
(3√2 + √2)
0]
[R] x [A] = [-4√2
4√2
0]
Resultant Vector
The resultant vector is given by the components of the matrix:
Resultant vector = -4√2 hat i + 4√2 hat j + 0 hat k
Simplifying further, we get:
Resultant vector = -4√2 hat i + 4√2 hat j
The correct option is (c) (11√2)/(√2) hat i + (1√2)/(√2) hat j, which can be simplified as 11 hat i + hat j.
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20.The vector vec A=5hat i 6hat j is rotated through an angle 45^(@) about the z-axis in the anticlockwise direction.The resultant vector is H.C.U.-2016 (a) 11hat i 1hat j (b) ihat i 11hat j (c) (11)/(sqrt(2))hat i (1)/(sqrt(2))hat j?
Question Description
20.The vector vec A=5hat i 6hat j is rotated through an angle 45^(@) about the z-axis in the anticlockwise direction.The resultant vector is H.C.U.-2016 (a) 11hat i 1hat j (b) ihat i 11hat j (c) (11)/(sqrt(2))hat i (1)/(sqrt(2))hat j? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about 20.The vector vec A=5hat i 6hat j is rotated through an angle 45^(@) about the z-axis in the anticlockwise direction.The resultant vector is H.C.U.-2016 (a) 11hat i 1hat j (b) ihat i 11hat j (c) (11)/(sqrt(2))hat i (1)/(sqrt(2))hat j? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 20.The vector vec A=5hat i 6hat j is rotated through an angle 45^(@) about the z-axis in the anticlockwise direction.The resultant vector is H.C.U.-2016 (a) 11hat i 1hat j (b) ihat i 11hat j (c) (11)/(sqrt(2))hat i (1)/(sqrt(2))hat j?.
20.The vector vec A=5hat i 6hat j is rotated through an angle 45^(@) about the z-axis in the anticlockwise direction.The resultant vector is H.C.U.-2016 (a) 11hat i 1hat j (b) ihat i 11hat j (c) (11)/(sqrt(2))hat i (1)/(sqrt(2))hat j? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about 20.The vector vec A=5hat i 6hat j is rotated through an angle 45^(@) about the z-axis in the anticlockwise direction.The resultant vector is H.C.U.-2016 (a) 11hat i 1hat j (b) ihat i 11hat j (c) (11)/(sqrt(2))hat i (1)/(sqrt(2))hat j? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 20.The vector vec A=5hat i 6hat j is rotated through an angle 45^(@) about the z-axis in the anticlockwise direction.The resultant vector is H.C.U.-2016 (a) 11hat i 1hat j (b) ihat i 11hat j (c) (11)/(sqrt(2))hat i (1)/(sqrt(2))hat j?.
Solutions for 20.The vector vec A=5hat i 6hat j is rotated through an angle 45^(@) about the z-axis in the anticlockwise direction.The resultant vector is H.C.U.-2016 (a) 11hat i 1hat j (b) ihat i 11hat j (c) (11)/(sqrt(2))hat i (1)/(sqrt(2))hat j? in English & in Hindi are available as part of our courses for Physics.
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20.The vector vec A=5hat i 6hat j is rotated through an angle 45^(@) about the z-axis in the anticlockwise direction.The resultant vector is H.C.U.-2016 (a) 11hat i 1hat j (b) ihat i 11hat j (c) (11)/(sqrt(2))hat i (1)/(sqrt(2))hat j?, a detailed solution for 20.The vector vec A=5hat i 6hat j is rotated through an angle 45^(@) about the z-axis in the anticlockwise direction.The resultant vector is H.C.U.-2016 (a) 11hat i 1hat j (b) ihat i 11hat j (c) (11)/(sqrt(2))hat i (1)/(sqrt(2))hat j? has been provided alongside types of 20.The vector vec A=5hat i 6hat j is rotated through an angle 45^(@) about the z-axis in the anticlockwise direction.The resultant vector is H.C.U.-2016 (a) 11hat i 1hat j (b) ihat i 11hat j (c) (11)/(sqrt(2))hat i (1)/(sqrt(2))hat j? theory, EduRev gives you an
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