Introduction to Electrodynamics · Problem 1.8

Vector Analysis — David J. Griffiths

Physics · how-vectors-transform

Problem

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

\overline{A}_y \overline{B}_y + \overline{A}_z \overline{B}_z = A_y B_y + A_z B_z

.) (b) What constraints must the elements (

R_{ij}

) of the three-dimensional rotation matrix (Eq. 1.30) satisfy, in order to preserve the length of

\mathbf{A}

(for all vectors

\mathbf{A}