The image explains the concept of rotation in the plane for coordinates in a two-dimensional space. It goes through the transformation of a point's coordinates from one coordinate system to a rotated one. Here’s a step-by-step explanation of the content:

1. Original Position of Point $P$

The point $P$ is initially described by its position vector $\vec{r}_P$ in the original (unrotated) coordinate system as: $\vec{r}_P = \begin{pmatrix} x_P \\ y_P \end{pmatrix}$ where $x_P$ and $y_P$ represent the $x$- and $y$-coordinates of $P$ in this original coordinate system.

2. Coordinates in the Rotated System

After rotating the coordinate system by an angle $\theta$, the point $P$ now has coordinates $x'_P$ and $y'_P$ in the rotated system. The new position vector $\vec{r}'_P$ is: $\vec{r}'_P = \begin{pmatrix} x'_P \\ y'_P \end{pmatrix}$

3. Transformation Equations

The coordinates $x'_P$ and $y'_P$ in the rotated system can be expressed in terms of the original coordinates $x_P$ and $y_P$. Using rotation formulas, we get: $x'_P = x_P \cos \theta + y_P \sin \theta$ $y'_P = y_P \cos \theta - x_P \sin \theta$ These equations show how the coordinates transform under a rotation by an angle $\theta$ in a counter-clockwise direction.

4. Solving for Original Coordinates $x_P$ and $y_P$

To express $x_P$ and $y_P$ in terms of $x'_P$ and $y'_P$, we rearrange the above equations: $x_P = x'_P \cos \theta - y'_P \sin \theta$ $y_P = y'_P \cos \theta + x'_P \sin \theta$ These are the inverse transformations, allowing you to convert back from the rotated coordinates to the original coordinates.

5. Representation with Unit Vectors

Finally, the position vector $\vec{r}_P$ is expressed in terms of unit vectors $\vec{e}_x$ and $\vec{e}_y$ in the original system and $\vec{e}'_x$ and $\vec{e}'_y$ in the rotated system: $\vec{r}_P = x_P \vec{e}_x + y_P \vec{e}_y = x'_P (\cos \theta \vec{e}_x + \sin \theta \vec{e}_y) + y'_P (\cos \theta \vec{e}_y - \sin \theta \vec{e}_x)$ This equation shows how the position vector can be decomposed in terms of the rotated basis vectors. It reaffirms that $\vec{r}_P$ can be represented in either the original or rotated coordinate system.

Summary of Key Points

• Coordinate Transformation: The coordinates transform between systems by rotation matrices.
• Inverse Transformation: We can recover original coordinates from rotated ones using inverse rotation.
• Vector Representation: The point's position can be expressed in terms of rotated or original basis vectors.

Would you like further elaboration on any specific part of this transformation?

Related Questions:

1. How do these transformations relate to rotation matrices in linear algebra?
2. What are the applications of such coordinate transformations in physics and engineering?
3. Could you explain how the transformation equations are derived?
4. How does this rotation affect the length of the position vector $\vec{r}_P$?
5. What is the significance of basis vectors $\vec{e}_x$ and $\vec{e}_y$ in these transformations?

Tip:

Remember that rotation transformations in a plane preserve the distance between points and the orientation (angle) relative to each other, which makes them useful in applications requiring rigid transformations, like in computer graphics and robotics.