Problem 5 - Precalculus

Question

The \(x^{\prime} y^{\prime}\) -coordinate system has been rotated \(\theta\) degrees from the \(x y\) -coordinate system. The coordinates of a point in the \(x y\) -coordinate system are given. Find the coordinates of the point in the rotated coordinate system. $$\theta=90^{\circ},(0,3)$$

Step-by-Step Solution

Answer

The coordinates of the point in the rotated system are (-3,0).

Step 1: Understand the Rotation Matrix

First, it should be understood that the rotation of coordinates can be achieved by using the rotation matrix. This matrix is given by \[\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right]\] where \(\theta\) is the angle by which the coordinates are rotated. In this exercise, the rotation angle is 90 degrees. So, first convert the angle into radians because the sine and cosine function in the rotation matrix is defined in radians. As \[90^\circ = \frac{\pi}{2}\] radians, now plug this value into the rotation matrix.

Step 2: Apply the Rotation Matrix to the Given Point

With \(\theta=\frac{\pi}{2}\), the rotation matrix becomes \[\left[\begin{array}{cc}\cos \frac{\pi}{2} & -\sin \frac{\pi}{2} \\ \sin \frac{\pi}{2} & \cos \frac{\pi}{2}\end{array}\right] = \left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]\] Now, apply this matrix to the given point (0,3) in the \(x,y\) coordinate system. This is done by matrix multiplication: \[(new x', new y') = \left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right] \cdot \left[\begin{array}{l} 0 \\ 3\end{array}\right]\]

Step 3: Calculate the New Coordinates

From the multiplication process, get the new x' and y' as follows: \[new x' = 0 \cdot 0 - 1 \cdot 3=-3;\] \[new y' = 1 \cdot 0 + 0 \cdot 3=0.\] This result means that the coordinates of the point in the rotated system are (-3,0).

Key Concepts

Rotation Matrix

In understanding how coordinate systems are manipulated, one key tool is the rotation matrix. A rotation matrix is used to rotate points in a coordinate system around the origin (0,0). For a given angle \( \theta \) in two-dimensional space, the rotation matrix is: \[\left[\begin{array}{cc}\cos \theta & -\sin \theta \ \sin \theta & \cos \theta\end{array}\right]\].

By multiplying this matrix by a column vector representing a point's coordinates, you can calculate the point's coordinates after rotation. This mathematical operation keeps the original distances and angles consistent, which is vital for many applications such as graphics programming, navigation, and robotics. To use this matrix correctly, ensure that the angle is in radian measure, which is a standard requirement for trigonometric functions in mathematics.

Coordinate Transformation

Coordinate transformation is a process that changes the viewpoint or reference frame for a given set of points. For instance, if you're moving from an \(xy\)-coordinate system to an \(x'y'\)-coordinate system through rotation about the origin, you're performing a coordinate transformation.

This transformation helps in various fields such as physics, engineering, and computer graphics, by altering the perspective of observation or interaction with objects and their relative positions. For example, when rotating a model in a 3D program, you're using coordinate transformation to display the model from a different angle. This is an essential concept to grasp, as it enables the translation of positions from one context to another, providing versatility and deeper understanding in spatial analyses.

Radian Angle Measure

Radian is the standard unit of angular measure used in many areas of mathematics. One radian is equivalent to the angle made at the center of a circle by an arc whose length is equal to the radius of the circle. In terms of degrees, one radian is approximately 57.2958 degrees. For the conversion, the formula is \( \theta_{radians} = \frac{\pi}{180} \cdot \theta_{degrees} \).

Understanding and using radians is crucial when working with trigonometric functions, which are defined in terms of radians. For example, the values of \(\sin\) and \(\cos\) functions are found using radian measures, as seen in the rotation matrix. Radian measure simplifies many mathematical expressions and is the preferred unit for angle measure in calculus and most higher mathematics, making it a fundamental concept in mathematics education.