[NOTES/ME-02001] Rotation of Coordinate Axes

Let $K$ be a set of coordinate axes $OX_1,OX_2,OX_3.$ Let ${K}^\prime$ be another set obtained from $K$ by applying a rotation by an angle $\theta$ about an axis passing through the origin. Let the unit vector along the axis of rotation be given by $\hat{n}$. A positive rotation is given by the right hand thumb rule. Hold the axis in right hand, with the thumb pointing along the unit vector $\hat{n}$. The positive direction of rotations is the one in which the fingers curl
 

Let the coordinates of a point be $(x_1,x_2,x_3)$ and $({x_1}^\prime,{x_2}^\prime,{x_3^\prime})$ w.r.t. the two sets of axes $K$ and $\acute{K}$. Then the components of the position vector are related by \begin{equation} {x}^\prime=R_{\hat{n}}(\theta)x, \end{equation} where \begin{eqnarray} x&=&\left[\begin{array}{c} x_1\\ x_2\\ x_3 \end{array}\right] \quad \text{~and~}\quad \acute{x}=\left[\begin{array}{c} \acute{x}_1\\\acute{x}_2\\ \acute{x}_3\end{array}\right] \end{eqnarray} and $R_{\hat{n}}(\theta)$ is a $3 \times 3$ matrix and will be called rotation matrix. For rotations about the coordinate axes we have the rotation matrices. \begin{equation} R_{1}(\alpha)=\left[\begin{array}{ccc} 1 &0 &0\\ 0 &\cos\alpha &\sin\alpha\\ 0 &-\sin\alpha &\cos\alpha \end{array}\right] \end{equation} \begin{equation} R_{2}(\alpha)=\left[\begin{array}{ccc} \cos\alpha &0 &-\sin\alpha\\ 0 &1 &0\\ \sin\alpha &0 &\cos\alpha \end{array}\right] \end{equation} \begin{equation} R_{3}(\alpha)=\left[\begin{array}{ccc} \cos\alpha &\sin\alpha &0\\ -\sin\alpha &\cos\alpha &0\\ 0 &0 &1 \end{array}\right] . \end{equation} 

Remarks

1. We will use a vector notation to collectively denote the components of position vector w.r.t. a chosen set of axis. So for example we will have \begin{equation} \vec{x} = (x_1,x_2,x_3), \quad \text{and} \quad \vec{x}^{\,\prime}=({x_1}^\prime,{x_2}^\prime,{x_3^\prime}). \end{equation}
2. For a point on the axis of rotation, the components of the position vector of the point do not change. So \begin{equation} \acute{x}=R_{\hat{n}}(\theta)x =x, \end{equation} if $x$ lies on the axis of rotation.
3. A unit vector $\hat{n}=(n_1,n_2,n_3)$ specifies the axis of rotation, $\theta$ gives the angle of rotation. Therefore, the number of independent parameters needed to specify a rotation is there because the parameters $n1,n_2,n_3$ satisfy a relation ${n_1}^2+{n_2}^2+{n_3}^2=1$.
4. Instead of rotating coordinate axes by and angle $\theta$, {\it one may rotate the vectors by an angle $-\theta$ and keep the axes fixed}. The components of the 'new' vector $\acute{\vec{A}}$, obtained by rotating a vector $\vec{A}$, will be related by the matrix $R_{\hat{n}}(-\theta)$ \begin{equation} \acute{A}=R_{\hat{n}}(-\theta)A \end{equation}
5. A rotation takes right(left) handed system of coordinate axis to another right(left) handed coordinate axes. The converse of this statement is also true. Any two right(left) handed systems of coordinate axes $K_1$ and $K_2$ are related by a rotation about some axis.

An explicit form of rotation matrix: Let $K$ and $\acute{K}$ be two sets of coordinate axes such that

• $l_1,m_1,n_1$ are direction cosines of the old $X_1$ axis w.r.t. the new axes $K^{\,\prime}$.
• $l_2,m_2,n_2$ are direction cosines of the old $X_2$ axis w.r.t. the new axes $K^{\,\prime}$.
• $l_3,m_3,n_3$ are direction cosines of the old $X_3$ axis w.r.t. the new axes $K^{\,\prime}$

For a point at unit distance from the origin and lying on the old axis $0{X_1}$, we have \begin{equation} x=\left[\begin{array}{c}1\\0\\0\end{array}\right] \text{ and } \acute{x}= \left[\begin{array}{c} l_1\\m_1\\n_1\end{array}\right]  \end{equation}  \begin{equation} \Longrightarrow \left[\begin{array}{c}l_1\\m_1\\n_1 \end{array}\right]=R\left[\begin{array}{c}1\\0\\0 \end{array}\right]. \end{equation} Similarly, by considering points on $0{X_2}$ and $0{X_3}$ we get \begin{equation} \left[\begin{array}{c}l_2\\m_2\\n_2\end{array}\right] =R \left[\begin{array}{c}0\\1\\0\end{array}\right], \qquad \left[\begin{array}{c}l_3\\m_3\\n_3\end{array}\right] =R\left[\begin{array}{c}0\\0\\1\end{array}\right]. \end{equation} Combining (9)-(10) we can write \begin{equation} R=\left[\begin{array}{clcr} l_1 &l_2 &l_3\\ m_1 &m_2 &m_3\\ n_1 &n_2 &n_3 \end{array}\right] . \end{equation}

Properties of rotation matrix

1. The matrix $R$ as obtained above has the following properties which may be verified directly using the properties of direction cosines..The matrix $R$ is an orthogonal matrix \begin{equation} R^{T}R=R R^{T}=I \Longrightarrow R^{-1}=R^T. \end{equation} Here $R^T$ denotes the transpose of the matrix $R$. Also $\det R=\vec{e}_1\cdot(\vec{e}_2\times\vec{e}_3)$ is the triple product $\vec{e}_1\cdot(\vec{e}_2\times\vec{e}_3)$ along the old axes, using their components in $K^{\,\prime}$. Since the axes $K$ and $\acute{K}$ are right handed, \begin{equation} \det R = \vec{e}_1\cdot(\vec{e}_2\times\vec{e}_3)=1. \end{equation} A matrix $O$ is called an orthogonal matrix if $O^{T}=O^{-1}$. Thus every rotation may be represented by an orthgonal matrix $R$ with $\det R=+1$. A three by three orthogonal matrix will have determinant $\pm1$. The rotations correspond to the case $\det R=1$. The case of $\det R=-1$ corresponds to a rotation followed by an inversion $\vec{x}^{\,\prime} = \vec{x}$.
2. The orthogonality property $R^{T} R=I$, $R\cdot R^{T}=I$ may also be written as \begin{eqnarray} \sum_{k} R_{i k} R_{j k}=\delta_{i j}, \qquad \sum_{k} R_{k i} R_{k j}=\delta_{i j} . \label{Eq13} \end{eqnarray}
3. The components of a vector $\vec{A}$ in two different frames are related by \begin{equation} \left[\begin{array}{c} \acute{A_1}\\ \acute{A_2}\\ \acute{A_3} \end{array}\right]=R\left[\begin{array}{c} {A_1}\\{A_2}\\{A_3} \end{array}\right] \end{equation} which will also be written as $\acute{A}=R A$, or, \begin{equation} \acute A_{l}=\sum R_{lm} A_m \label{Eq15} \end{equation}
4. If we have two vectors $\vec{A}$ and $\vec{B}$, their components will depend on the choice of coordinate axes. But the scalar product $\vec{A} \cdot \vec{B}$ computed from $A_1 B_1+A_2 B_2+A_3 B_3$ is the same as computed from $\acute{A_1}\acute{B_1}+\acute{A_2}\acute{B_2}+\acute{A_3}\acute{B_3}$ To prove the above statement we consider \begin{eqnarray} \sum_k\acute{A_k}\acute{B_k} &=& \sum_k\sum_i (R_{k i} A_i)\sum_j (R_{k j} B_j)= \sum_{i,j}\sum_k(R_{k i}R_{k j})A_i B_j \label{Eq17}\\ &=&\sum_{i,j}\delta_{ij} A_{i}B_{j}= \sum_i A_iB_i, \label{Eq18} \end{eqnarray} \begin{equation} \therefore \qquad \acute{A_1}\acute{B_1}+\acute{A_2}\acute{B_2}+\acute{A_3}\acute{B_3} =\sum_i A_{i}B_{i}= A_1 B_1+A_2 B_2+A_3 B_3, \end{equation} where in the step from \EqRef{EQ17} to \EqRef{EQ18} the orthogonality property \EqRef{EQ13} has been used. The above result can be stated as a property of rotations that under a rotation scalar product of two vectors remains invariant. This implies that the length of a vector $\Vert A\Vert( =\sqrt{(A\cdot A)}$, and the angle between two vectors, \begin{equation} \cos\theta=\frac{\vec{A}\cdot \vec{B}}{\Vert\vec{A}\Vert\, \Vert\vec{B}\Vert}, \end{equation} are also invariant. Similarly, it can be proved that the volume of a parallelopiped with edges, given by $\vec{A},\vec{B},\vec{C}$,  does not change under rotation. These properties make use of the fact that the rotations are represented by orthogonal matrices with determinant +1.
5. Conversely, every orthogonal matrix $R$ with $\det R=+1$ represents a rotation. Knowing $R$ and a set of axis $K$, the new set of axes $\acute{K}$ is easily found; the three columns of $R$ give the components of unit vectors along the three new axis.
6. Let $S$ be a $3 \times 3$ matrix such that $ \sum {\acute{A_k}}^2=\sum {A_k}^2$ holds for every vector $\vec{A}$, where \begin{equation} \left[\begin{array}{c} \acute{A_1}\\ \acute{A_2}\\ \acute{A_3} \end{array}\right]= S \begin{bmatrix}A_1\\A_2\\A_3 \end{bmatrix} . \end{equation} In other words, if the length of a vector $\vec{A}$ as computed from its components $(A_1,A_2,A_3)$ is equal to that computed from the transformed components $(\acute{A_1},\acute{A_2},\acute{A_3})$, then $S$ is an orthogonal matrix: \begin{equation} S^{T} S=I.\label{Eq22} \end{equation} The equality $\det S=1$ does not follow from \EqRef{EQ22}, it implies only $\det S=\pm 1$.
7. Since a rotation matrix $R$ is $3\times3$ orthogonal matrix with determinant 1, the eigenvalues of $R$ are given by $1, e^{i\theta}$, $e^{-i\theta}$. The eigenvector corresponding to $1$ will give the axis of rotation.