Chapter 3: Q5P (page 158)

Show that the $C$ matrix in (11.10) does represent a rotation by finding the rotation angle. Write equations (7.13) and (11.13) for this rotation.

Short Answer

Expert verified

$x = x^{'} \cos 63.43 - y^{'} \sin 63.43 y = x^{'} \sin 63.43 - y^{'} \cos 63.43$

Step by step solution

Given information

Matrix $C = (\begin{matrix} \frac{1}{\sqrt{5}} \\ \frac{2}{\sqrt{5}} \end{matrix} \begin{matrix} \frac{- 2}{\sqrt{5}} \\ \frac{1}{\sqrt{5}} \end{matrix})$

Determinant of a matrix

The signed factor by which this matrix scales regions is called the determinant of a matrix. If the sign is negative, the matrix's orientation is reversed.

Calculate the determinant of C

The determinant of $C$ .

$d e t C = |\begin{matrix} \frac{1}{\sqrt{5}} & \frac{- 2}{\sqrt{5}} \\ \frac{2}{\sqrt{5}} & \frac{1}{\sqrt{5}} \end{matrix}| = \frac{1}{5} - (- \frac{4}{5}) = 1$

Matrix $C$ represents a rotation.

Equate matrix $C$ by the general rotation matrix.

$\cos θ = \frac{1}{\sqrt{5}} θ = 63.43 °$

Substitute the value of θ

Substitute the value of $θ$ .

$(\begin{matrix} X \\ Y \end{matrix}) = (\begin{matrix} \cos 63.43 \\ \sin 63.43 \end{matrix} \begin{matrix} - \sin 63.43 \\ \cos 63.43 \end{matrix}) (\begin{matrix} x \\ y \end{matrix})$

This gives,

$x = x^{'} \cos 63.43 - y^{'} \sin 63.43 y = x^{'} \sin 63.43 - y^{'} \cos 63.43$

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Show that the $C$ matrix in (11.10) does represent a rotation by finding the rotation angle. Write equations (7.13) and (11.13) for this rotation.

Short Answer

Step by step solution

Given information

Determinant of a matrix

Calculate the determinant of C

Substitute the value of θ

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Show that the C matrix in (11.10) does represent a rotation by finding the rotation angle. Write equations (7.13) and (11.13) for this rotation.

Short Answer

Step by step solution

Given information

Determinant of a matrix

Calculate the determinant of C

Substitute the value of θ

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Translational Dynamics

Geometrical and Physical Optics

Famous Physicists

Further Mechanics and Thermal Physics

Measurements

Fundamentals of Physics

Study anywhere. Anytime. Across all devices.

Show that the $C$ matrix in (11.10) does represent a rotation by finding the rotation angle. Write equations (7.13) and (11.13) for this rotation.