Chapter 3: Q30P (page 131)

For the matrices Gand K in $(7.21)$ , find the matrices $R = GK$ and $S = KG$ . Note that $R \neq S$ . (In3dimensions, rotations about two different axes do not in general commute.) Find what geometric transformations are produced byRand, S.

Short Answer

Expert verified

The geometric transformations are that the rotation axis of R its z axis and the rotation angle is $90^{\circ}$ and that of S is x axis and the rotation angle is $90^{\circ}$

Step by step solution

Matrix Transformations

Matrix transformation can be of two types: rotation and reflection only for square matrices. When the determinant value of the matrix is1then it is termed rotation and if the value is -1 then, it is a reflection.

Given Parameters

$G = (\begin{matrix} 0 & 0 & 1 \\ 0 & - 1 & 0 \\ 1 & 0 & 0 \end{matrix}) and K = (\begin{matrix} 0 & 0 & 1 \\ - 1 & 0 & 0 \\ 0 & - 1 & 0 \end{matrix})$

Calculating R and S

$R = GK = (\begin{matrix} 0 & 0 & 1 \\ 0 & - 1 & 0 \\ 1 & 0 & 0 \end{matrix}) (\begin{matrix} 0 & 0 & 1 \\ - 1 & 0 & 0 \\ 0 & - 1 & 0 \end{matrix}) = (\begin{matrix} 0 & - 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}) S = KG = (\begin{matrix} 0 & 0 & 1 \\ - 1 & 0 & 0 \\ 0 & - 1 & 0 \end{matrix}) (\begin{matrix} 0 & 0 & 1 \\ 0 & - 1 & 0 \\ 1 & 0 & 0 \end{matrix}) = (\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & - 1 \\ 0 & 1 & 0 \end{matrix})$

Therefore, $R \neq S$ and G ,K do not commute.

Determinants of R and S need to be calculated.

$d e t R = 1$ and $d e t S = 1$ hence both matrices are rotations.

The rotation axis of R its z axis and the rotation angle is $90^{\circ}$ and that of S is x axis and the rotation angle is $90^{\circ}$ .

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For the matrices Gand K in $(7.21)$ , find the matrices $R = GK$ and $S = KG$ . Note that $R \neq S$ . (In3dimensions, rotations about two different axes do not in general commute.) Find what geometric transformations are produced byRand, S.

Short Answer

Step by step solution

Matrix Transformations

Given Parameters

Calculating R and S

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For the matrices Gand K in(7.21), find the matricesR=GK andS=KG. Note thatR≠S. (In3dimensions, rotations about two different axes do not in general commute.) Find what geometric transformations are produced byRand, S.

Short Answer

Step by step solution

Matrix Transformations

Given Parameters

Calculating R and S

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Famous Physicists

Electric Charge Field and Potential

Electricity

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Atoms and Radioactivity

Further Mechanics and Thermal Physics

Study anywhere. Anytime. Across all devices.

For the matrices Gand K in $(7.21)$ , find the matrices $R = GK$ and $S = KG$ . Note that $R \neq S$ . (In3dimensions, rotations about two different axes do not in general commute.) Find what geometric transformations are produced byRand, S.