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Chapter 10: Q4P (page 502)

Any rotation of axes in three dimensions can be described by giving the nine direction cosines of the angle between the (x,y,z)axes and the (x',y',z')axes. Show that the matrix A of these direction cosines in (2.7)or (2.10)is an orthogonal matrix. Hint: See Chapter 3, Section 9. Find AATand use Problem 3.

Short Answer

Expert verified

Answer

The statement is verified andAAT=I

Step by step solution

01

Given Information

Matrix A with direction cosinesx,y,z,x',y',z'.

02

Definition of Orthogonal matrix.

A Square matrix is said to be orthogonal matrix if AAT=I.

03

Show that A is orthogonal.

Matrix A with direction cosines x,y,z,x',y',z.

A-a,,a2,a3AT=a1Ta2Ta3T

The value of AATis as follows.

AAT=鈭�akiakjk=13AAI=aiIajAAT=未ij

Hence, the statement is verified and AAT=I.

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