(a) Prove that the two-dimensional rotation matrix (Eq.

Chapter 1, Problem 8P

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QUESTION:

(a) Prove that the two-dimensional rotation matrix (Eq. 1.29) preserves dot products. (That is, show that .)

(b) What constraints must the elements (Ri j) of the three-dimensional rotation matrix (Eq. 1.30) satisfy, in order to preserve the length of A (for all vectors A)?

Reference equation 1.30

,

Reference equation 1.29

.

Questions & Answers

QUESTION:

(a) Prove that the two-dimensional rotation matrix (Eq. 1.29) preserves dot products. (That is, show that .)

(b) What constraints must the elements (Ri j) of the three-dimensional rotation matrix (Eq. 1.30) satisfy, in order to preserve the length of A (for all vectors A)?

Reference equation 1.30

,

Reference equation 1.29

.

ANSWER:

Step 1 of 5

In part (a) we have to prove that  using given two dimensional rotation matrix equations.

Given the two dimensional rotation matrix is,

Using the above equations,

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