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91. Distances between points in a plane do not change when a coordinate system is rotated. In other words, the magnitude of a vector is invariant under rotations of the coordinate system. Suppose a coordinate system S is rotated about its origin by angle o to become a new yxp- xo)² + (Yp- y@² = Vcx°p – x'q}² + (v'p- y' q}². coordinate system S' , as shown in the following figure. A point in a plane has coordinates (x, y) in S and coordinates (*, у) in S'. S (a) Show that, during the transformation of rotation, the coordinates in S' are expressed in terms of the coordinates in S by the following relations: S' Sx =x cos o + y sin o ly = -x sin o + ycos q' (b) Show that the distance of point P to the origin is invariant under rotations of the coordinate system. Here, you have to show that ? + y² = \&°² + y?. (c) Show that the distance between points P and Q is invariant under rotations of the coordinate system. Here, you have to show that

91. Distances between points in a plane do not change when a coordinate system is rotated. In other words, the magnitude of a vector is invariant under rotations of the coordinate system. Suppose a coordinate system S is rotated about its origin by angle o to become a new yxp- xo)² + (Yp- y@² = Vcx°p – x'q}² + (v'p- y' q}². coordinate system S' , as shown in the following figure. A point in a plane has coordinates (x, y) in S and coordinates (*, у) in S'. S (a) Show that, during the transformation of rotation, the coordinates in S' are expressed in terms of the coordinates in S by the following relations: S' Sx =x cos o + y sin o ly = -x sin o + ycos q' (b) Show that the distance of point P to the origin is invariant under rotations of the coordinate system. Here, you have to show that ? + y² = \&°² + y?. (c) Show that the distance between points P and Q is invariant under rotations of the coordinate system. Here, you have to show that

Physics for Scientists and Engineers
Physics for Scientists and Engineers
10th Edition
ISBN:9781337553278
Author:Raymond A. Serway, John W. Jewett
Publisher:Raymond A. Serway, John W. Jewett
Chapter13: Universal Gravitation
Section: Chapter Questions
Problem 32AP
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91. Distances between points in a plane do not change when a coordinate system is rotated. In other words, the magnitude of a vector is invariant under rotations of the coordinate system. Suppose a coordinate system S is rotated about its origin by angle o to become a new yxp- xo)² + (Yp- y@² = Vcx°p – x'q}² + (v'p- y' q}². coordinate system S' , as shown in the following figure. A point in a plane has coordinates (x, y) in S and coordinates (*, у) in S'. S (a) Show that, during the transformation of rotation, the coordinates in S' are expressed in terms of the coordinates in S by the following relations: S' Sx =x cos o + y sin o ly = -x sin o + ycos q' (b) Show that the distance of point P to the origin is invariant under rotations of the coordinate system. Here, you have to show that ? + y² = \&°² + y?. (c) Show that the distance between points P and Q is invariant under rotations of the coordinate system. Here, you have to show that
Transcribed Image Text:91. Distances between points in a plane do not change when a coordinate system is rotated. In other words, the magnitude of a vector is invariant under rotations of the coordinate system. Suppose a coordinate system S is rotated about its origin by angle o to become a new yxp- xo)² + (Yp- y@² = Vcx°p – x'q}² + (v'p- y' q}². coordinate system S' , as shown in the following figure. A point in a plane has coordinates (x, y) in S and coordinates (*, у) in S'. S (a) Show that, during the transformation of rotation, the coordinates in S' are expressed in terms of the coordinates in S by the following relations: S' Sx =x cos o + y sin o ly = -x sin o + ycos q' (b) Show that the distance of point P to the origin is invariant under rotations of the coordinate system. Here, you have to show that ? + y² = \&°² + y?. (c) Show that the distance between points P and Q is invariant under rotations of the coordinate system. Here, you have to show that
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