A book lies face up on a table

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Linear algebra and vector spaces

Mathematical Methods for Physics, Ch Matrices, determinants and systems of linear equations

Problem

A book lies face up on a table. Take

x

east,

y

north and

z

up. A quarter turn of the book about the

x

axis leaves

\hat{\mathbf{x}}

alone and sends

\hat{\mathbf{y}}

\hat{\mathbf{z}}

and

\hat{\mathbf{z}}

-\hat{\mathbf{y}}

. A quarter turn about the

z

axis leaves

\hat{\mathbf{z}}

alone and sends

\hat{\mathbf{x}}

\hat{\mathbf{y}}

and

\hat{\mathbf{y}}

-\hat{\mathbf{x}}

. ((a)) Write the two turns as three-by-three matrices

\mathsf{R}_x

and

\mathsf{R}_z

. ((b)) Compute

\mathsf{R}_z\mathsf{R}_x

and

\mathsf{R}_x\mathsf{R}_z

, and say which of them means turning about

x

first and then about

z

. ((c)) The front cover's outward direction starts along

\hat{\mathbf{z}}

. Find where it points after each order of the two turns, and explain in one sentence why the orders differ. ((d)) Evaluate the determinant of each product, and explain why the two determinants are equal although the products are not.

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