Chapter 2, Problem 91CP

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 $\phi$ to become a new coordinate system S’, as shown in the following figure. A point in a plane has coordinates $\left(x,y\right)$in S and coordinates $\left(x\text{'},y\text{'}\right)$ in 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: $\{\begin{array}{c}x\text{'}=x\cos \phi +y\sin \phi \\ y\text{'}=-x\sin \phi +y\cos \phi \end{array}$

(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 $\sqrt{x^2+y^2}=\sqrt{x{\text{'}}^2+y{\text{'}}^2}$.

(c) Show that the distance between points $P$and $Q$ is invariant under rotations of the coordinate system. Here, you have to show that $\sqrt{{\left(x_P=x_Q\right)}^2+{\left(y_P-y_Q\right)}^2}=\sqrt{{\left(x{\text{'}}_P=x{\text{'}}_Q\right)}^2+{\left(y{\text{'}}_P-y{\text{'}}_Q\right)}^2}$ .