Chapter 3: Q4P (page 158)

Find the inverse of the rotation matrix in (7.13); you should get $C$ in (11.14). Replace role="math" localid="1664340540940" $θ$ by $- θ$ in (7.13) to see that the matrix C corresponds to a rotation through $- θ$ .

Short Answer

Expert verified

Inverse is $(\begin{matrix} \cos θ \\ \sin θ \end{matrix} \begin{matrix} - \sin θ \\ \cos θ \end{matrix})$

Step by step solution

Given information

The rotation matrix is $(\begin{matrix} \cos θ \\ - \sin θ \end{matrix} \begin{matrix} \sin θ \\ \cos θ \end{matrix})$

Rotation matrix

A rotation matrix is a transformation matrix that operates on a vector and outputs a rotated vector while keeping the coordinate axes constant. These matrices rotate a vector by an angle in a counter clockwise direction. A square matrix with real entities is always a rotation matrix.

Calculate the inverse of the rotation matrix

Replace $θ$ by $- θ$ in the given rotation matrix.

${(\begin{matrix} \cos θ \\ - \sin θ \end{matrix} \begin{matrix} \sin θ \\ \cos θ \end{matrix})}^{- 1} = (\begin{matrix} \cos (- θ) \\ - \sin (- θ) \end{matrix} \begin{matrix} \sin (- θ) \\ \cos (- θ) \end{matrix}) = (\begin{matrix} \cos θ \\ \sin θ \end{matrix} \begin{matrix} - \sin θ \\ \cos θ \end{matrix})$

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Find the inverse of the rotation matrix in (7.13); you should get $C$ in (11.14). Replace role="math" localid="1664340540940" $θ$ by $- θ$ in (7.13) to see that the matrix C corresponds to a rotation through $- θ$ .

Short Answer

Step by step solution

Given information

Rotation matrix

Calculate the inverse of the rotation matrix

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Find the inverse of the rotation matrix in (7.13); you should get C in (11.14). Replace role="math" localid="1664340540940" θ by -θ in (7.13) to see that the matrix C corresponds to a rotation through -θ.

Short Answer

Step by step solution

Given information

Rotation matrix

Calculate the inverse of the rotation matrix

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Further Mechanics and Thermal Physics

Torque and Rotational Motion

Wave Optics

Solid State Physics

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Magnetism and Electromagnetic Induction

Study anywhere. Anytime. Across all devices.

Find the inverse of the rotation matrix in (7.13); you should get $C$ in (11.14). Replace role="math" localid="1664340540940" $θ$ by $- θ$ in (7.13) to see that the matrix C corresponds to a rotation through $- θ$ .