Q6P Write the transformation equatio... [FREE SOLUTION]

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Chapter 10: Q6P (page 517)

Write the transformation equations to show that $鈭� 脳 V$ is a pseudo vector if Vis a vector. Hint:See equations (5.13), (6.2), and (6.3).

Short Answer

Expert verified

The transformation equation is ${(鈭� 脳 V)_{伪}}^{'} = (d e t A) a_{伪} i (鈭� 脳 V)_{i}$

Step by step solution

Given information.

Matrix definitions are given.

Definition of a rotation matrix.

The rotation matrix is defined in this way.

$[\begin{matrix} c o s 蠒 & - s i n 蠒 \\ s i n 蠒 & c o s 蠒 \end{matrix}]$

Define a vector and an orthogonal matrix.

Define to be a vector. Define an orthogonal matrix denoting proper or improper rotation. Write the result for proper and improper rotations.

$\frac{鈭�}{鈭� {x_{i}}^{'}} = a_{i} j \frac{鈭�}{鈭� x_{j}}$

Continue evaluations.

Continue with evaluations.

${(鈭� 脳 V)}_{伪}' = 蔚'_{伪尾纬} \frac{鈭�}{鈭� X_{尾}'} V_{纬}' = (d e t A) a_{伪 i} a_{尾 j} a_{纬 k} 蔚_{i j k} a_{尾 m} \frac{鈭�}{鈭� X_{m}} a_{纬 n} v_{n} = (d e t A) a_{伪 i} a_{尾 j} a_{尾 m} a_{纬 k} a_{纬 n} 蔚_{i j k} \frac{鈭�}{鈭� X_{m}} v_{n}$

Continue the simplification.

${(鈭� 脳 V)}_{伪}' = (d e t A) a_{伪 i} {鈭�}_{j m} {鈭�}_{k n} 蔚_{i j k} \frac{鈭�}{鈭� X_{m}} v_{n} = (d e t A) a_{伪 i} 蔚_{i j k} \frac{鈭�}{鈭� X_{m}} v_{k} = (d e t A) a_{伪 i} {(鈭� 脳 V)}_{i}$

This implies that $鈭� 脳 V$ is a pseudo vector. Since A is defined to be orthogonal, write its result.

$a_{尾} j a_{尾} m = 未_{j} m A^{T} A = I$

Therefore, write the result.

${(鈭� 脳 V)_{伪}}^{'} = (d e t A) a_{伪} i (鈭� 脳 V)_{i}$

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91影视

Write the transformation equations to show that $鈭� 脳 V$ is a pseudo vector if Vis a vector. Hint:See equations (5.13), (6.2), and (6.3).

Short Answer

Step by step solution

Given information.

Definition of a rotation matrix.

Define a vector and an orthogonal matrix.

Continue evaluations.

One App. One Place for Learning.

Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

91影视

Write the transformation equations to show that 鈭�脳Vis a pseudo vector if Vis a vector. Hint:See equations (5.13), (6.2), and (6.3).

Short Answer

Step by step solution

Given information.

Definition of a rotation matrix.

Define a vector and an orthogonal matrix.

Continue evaluations.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Energy Physics

Wave Optics

Quantum Physics

Electricity

Kinematics Physics

Space Physics

Study anywhere. Anytime. Across all devices.

Write the transformation equations to show that $鈭� 脳 V$ is a pseudo vector if Vis a vector. Hint:See equations (5.13), (6.2), and (6.3).