A vector v = (x, y, z) is rotated by an angle about an axis having unit vector (a, b

Chapter 10, Problem T2

(choose chapter or problem)

A vector v = (x, y, z) is rotated by an angle \(\theta\) about an axis having unit vector (a, b, c), thereby forming the rotated vector \(\mathbf{v}_{R}=\left(x_{R}, y_{R}, z_{R}\right)\). It can be shown that

\(\left[\begin{array}{l}
x_{R} \\
y_{R} \\
z_{R}
\end{array}\right]=R(\theta)\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]\)

with

\(\begin{aligned}
R(\theta)=\cos (\theta)\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]+ & +(1-\cos (\theta))\left[\begin{array}{l}
a \\
b \\
c
\end{array}\right]\left[\begin{array}{lll}
a & b & c
\end{array}\right] \\
& +\sin (\theta)\left[\begin{array}{rrr}
0 & -c & b \\
c & 0 & -a \\
-b & a & 0
\end{array}\right]
\end{aligned}\)

(a) Use a computer to show that \(R(\theta) R(\varphi)=R(\theta+\varphi)\), and then give a physical reason why this must be so. Depending on the sophistication of the computer you are using, you may have to experiment using different values of a, b, and

\(c=\sqrt{1-a^{2}-b^{2}}\)

(b) Show also that \(R^{-1}(\theta)=R(-\theta)\) and give a physical reason why this must be so.

(c) Use a computer to show that \(\operatorname{det}(R(\theta))=+1\).