Q3P For Example 1, write out the com... [FREE SOLUTION]

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Chapter 10: Q3P (page 517)

For Example 1, write out the components of U,V, and $U 脳 V$ in the original right-handed coordinate system and in the left-handed coordinate system S' with the axis reflected. Show that each component of $U 脳 V$ inS'has the 鈥渨rong鈥� sign to obey the vector transformation laws.

Short Answer

Expert verified

The expressions are proved below.

Step by step solution

Given information.

Matrix definitions are given.

Definition of a rotation matrix.

The rotation matrix is defined in this way.

$[\begin{matrix} 肠辞蝉蠒 & - \sin 蠒 \\ \sin 蠒 & \cos 蠒 \end{matrix}]$

Write relevant definitions.

Define an orthogonal matrix A and a reflection about the x -y plane. Define two coordinate systems S be the actual system and S' the system with the inverted axis on which vectors U,V are defined.

$A = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & - 1 \end{matrix}] U' (V') = [\begin{matrix} U_{1} (V_{1}) \\ U_{2}' (V_{2}') \\ U_{3}' (V_{3}') \end{matrix}] U' (V') = [\begin{matrix} U_{1} (V_{1}) \\ U_{2} (V_{2}) \\ - U_{3} (V_{3}) \end{matrix}]$

Take the cross-product of the two vectors.

Take the cross-product of the two vectors and continue the simplification.

$U' 脳 V' = [\begin{matrix} {\hat{e}}_{1} & {\hat{e}}_{2} & {\hat{e}}_{3} \\ U_{1} & U_{2} & - U_{3} \\ V_{1} & V_{2} & - V_{3} \end{matrix}] = [\begin{matrix} V_{2} U_{3} - V_{3} U_{2} \\ V_{3} U_{1} - V_{1} U_{3} \\ V_{2} U_{1} - V_{1} U_{2} \end{matrix}] = - [\begin{matrix} U_{2} V_{3} - U_{3} V_{2} \\ U_{3} V_{1} - U_{1} V_{3} \\ - (U_{1} V_{2} - U_{2} V_{1}) \end{matrix}]$

This results in $P = U 脳 V$ .

P is described by the following definition of vectors and pseudo vectors.

$V_{i}' = a_{ij} V_{j} P_{i}' = (\det A) a_{ij} P_{j}$

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91影视

For Example 1, write out the components of U,V, and $U 脳 V$ in the original right-handed coordinate system and in the left-handed coordinate system S' with the axis reflected. Show that each component of $U 脳 V$ inS'has the 鈥渨rong鈥� sign to obey the vector transformation laws.

Short Answer

Step by step solution

Given information.

Definition of a rotation matrix.

Write relevant definitions.

Take the cross-product of the two vectors.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

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91影视

For Example 1, write out the components of U,V, and U脳Vin the original right-handed coordinate system and in the left-handed coordinate system S' with the axis reflected. Show that each component ofU脳VinS'has the 鈥渨rong鈥� sign to obey the vector transformation laws.

Short Answer

Step by step solution

Given information.

Definition of a rotation matrix.

Write relevant definitions.

Take the cross-product of the two vectors.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Wave Optics

Oscillations

Electricity

Force

Rotational Dynamics

Further Mechanics and Thermal Physics

Study anywhere. Anytime. Across all devices.

For Example 1, write out the components of U,V, and $U 脳 V$ in the original right-handed coordinate system and in the left-handed coordinate system S' with the axis reflected. Show that each component of $U 脳 V$ inS'has the 鈥渨rong鈥� sign to obey the vector transformation laws.