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Chapter 2: Problem 3

Give the standard matrix representation of the rotation of the plane clockwise about the origin through an angle of a. \(45^{\circ}\), b. \(60^{\circ}\), c. \(150^{\circ}\).

Short Answer

Expert verified
The standard matrices are: a) \(\begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ -\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix}\), b) \(\begin{bmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & \frac{1}{2} \end{bmatrix}\), c) \(\begin{bmatrix} -\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix}\).

Step by step solution

01

Understand the Rotation Matrix

A rotation matrix in two dimensions for a rotation about the origin by an angle \( \theta \) is given by \[ R(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{bmatrix} \]. When dealing with a clockwise rotation, we must adjust the signs in our matrix.
02

Clockwise Rotation Matrix Formula

For a clockwise rotation, the standard matrix is adjusted by changing the signs of the sine components: \[ R_{clockwise}(\theta) = \begin{bmatrix} \cos(\theta) & \sin(\theta) \ -\sin(\theta) & \cos(\theta) \end{bmatrix} \].
03

Calculate the Matrix for 45掳

Substitute \( \theta = 45^{\circ} \) into the clockwise rotation formula. \[ R_{clockwise}(45^{\circ}) = \begin{bmatrix} \cos(45^{\circ}) & \sin(45^{\circ}) \ -\sin(45^{\circ}) & \cos(45^{\circ}) \end{bmatrix} = \begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \ -\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix} \].
04

Calculate the Matrix for 60掳

Substitute \( \theta = 60^{\circ} \) into the clockwise rotation formula. \[ R_{clockwise}(60^{\circ}) = \begin{bmatrix} \cos(60^{\circ}) & \sin(60^{\circ}) \ -\sin(60^{\circ}) & \cos(60^{\circ}) \end{bmatrix} = \begin{bmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \ -\frac{\sqrt{3}}{2} & \frac{1}{2} \end{bmatrix} \].
05

Calculate the Matrix for 150掳

Substitute \( \theta = 150^{\circ} \) into the clockwise rotation formula. \[ R_{clockwise}(150^{\circ}) = \begin{bmatrix} \cos(150^{\circ}) & \sin(150^{\circ}) \ -\sin(150^{\circ}) & \cos(150^{\circ}) \end{bmatrix} = \begin{bmatrix} -\frac{\sqrt{3}}{2} & \frac{1}{2} \ -\frac{1}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

clockwise rotation
To understand clockwise rotation, picture a clock hanging on a wall. The hands of the clock move in a circular path to the right, this direction is what we call clockwise. In mathematics, especially when working with rotation matrices, we need to adjust our calculations to match this direction. Typically, rotations are considered counterclockwise, so when we want a clockwise rotation, we need to change the signs of certain terms in our matrix representation. Essentially:
  • In the standard rotation matrix, adjust the sine terms to reflect a clockwise path by reversing their signs.
  • This adjustment ensures that your entire system rotates towards the right, mimicking the hands of a clock.
Recognizing the direction helps in correctly applying your rotation formula.
two-dimensional rotation
Two-dimensional rotation involves turning a shape or a system around a central point in a flat plane, like turning a piece of paper around a pin pushed through its center. Things to keep in mind:
  • The rotation happens around a specific point known as the axis of rotation, in 2D this is generally the origin (0,0) point.
  • Unlike in three-dimensional rotation, where there might be tipping or flipping, in 2D everything happens on a two-directional plane: width and height.
The outcome of a rotation in two dimensions is always the same shape, only in a different orientation relative to its original position.
matrix representation
Matrix representation in rotations allows us to transform points on a plane using linear algebra. Here's how it works:
  • The rotation matrix for a given angle \(\theta\) serves as the tool that calculates where every point in your system will move once rotated.
  • For a standard clockwise rotation, we use a 2x2 matrix that incorporates trigonometric functions鈥攃osine and sine鈥攚hich handle the rotation calculations.
This matrix translates the degree of rotation into precise numerical changes applied to the coordinates, ensuring the integrity of the shape's rotation is maintained.
angle of rotation
The angle of rotation is a fundamental concept defining how far a shape rotates around the center. A few key notes:
  • Angles are typically measured in degrees or radians.
  • The magnitude of the angle determines how much the shape has been turned from its starting point.
If you picture a pizza, the angle of rotation tells you how much you need to spin one slice around the center to move it to another position, either nearer to 0掳 (its starting point) or all the way around perhaps even back to its original spot. For practical use, understanding whether we use clockwise or counterclockwise angles can dictate the adjustment needed in mathematical terms, like in reversing sine terms in our rotation matrix for a clockwise turn.

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