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Chapter 3: Q7P (page 159)

Generalize Problem 6 to three dimensions; to n dimensions.

Short Answer

Expert verified

C is an orthogonal matrix.

Step by step solution

01

Given information

C is a matrix whose columns are the components x1,y1 and x2,y2 of two perpendicular vectors, each of unit length.

02

Orthogonal matrix

An orthogonal matrix, also known as an orthonormal matrix, is a real square matrix with orthonormal vectors in the columns and rows.

03

Calculate matrix  C

To generalize matrix C, we need to represent n-dimensional matrix. Therefore, we will need northogonal vectors rneach with ncomponents.

A n-dimensional matrix can be mathematically presented as

C=(r1r2r2.......rn),whereriTrj=未ij,and未ij=10i=ji鈮�j

Now, calculate CTC.

CTC=r1Tr2T鈰�..rnTr1r2r2.........rn

Therefore, CTCij=riTrj=未ij, which shows that C is an orthogonal matrix.

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