[NOTES/ME-02006]-Change of coordinate axes

Define two sets of axes with common origin

We know that every vector \({\mathbf A}\) can be expressed as linear combination of unit vectors along the three coordinate axes. The coefficients are called components of the vector. The components of a given vector will be different w.r.t. different coordinate systems. Here we present a way of relating the components of a vector w.r.t. two different sets of axes. Let us assume that we have two right handed coordinate systems, \(K,K {'}\), whose origins coincide but the axes are orientated differently. The coordinates of a point \(P\) as seen in two frames will be different. We wish to find relation between the components w.r.t. the two sets of axes.

Define direction cosines

Let \({\bf i,j,k}\) denote unit vectors along the axes \(K\) and \(\mathbf {l,m,n}\) denote unit vectors along the new axes. Let the components of unit vectors \(\mathbf {l,m,n}\) w.r.t. the old axes be written as \begin{eqnarray} \vec{\ell} = (\ell_1, \ell_2, \ell_3);\quad \vec{m} = (m_1, m_2, m_3); \quad \vec{n} &=& (m_1, m_2, m_3). \end{eqnarray} Then we have \begin{eqnarray}\label{EQ04} {\mathbf l} = \ell_1 {\bf i} + \ell_2 {\bf j} + \ell_3 {\bf k}; \qquad {\mathbf m} = m_1 {\bf i} + m_2 {\bf j} + m_3 {\bf k};\qquad {\mathbf n} = n_1 {\bf i} + n_2 {\bf j} + n_3 {\bf k} \end{eqnarray}

Get a vector as linear combinations in the two bases 

Let \((x,y,z)\equiv\vec{r}\) and \((x{'},y{'},z{'})\equiv\vec{r}{'}\) denote the components of the position vector \(\overrightarrow{OP}\) of \(P\) the w.r.t. the frames \(K,K{'}\). Thus we have \begin{eqnarray} {\bf r} &=& x {\bf i} + y {\bf j} + z {\bf k}\\ &=&x{'} {\bf l} + y{'} {\bf m} + z{'} {\bf n}\label{EQ06} \end{eqnarray} Using \eqRef{EQ04} we get \begin{eqnarray} {\bf r} &=&x{'} {\bf l} + y{'} {\bf m} + z{'} {\bf n}\\ &=& x{'} ( \ell_1 {\bf i} + \ell_2 {\bf j} + \ell_3 {\bf k}) + y{'} ( m_1 {\bf i} + m_2 {\bf j} + m_3 {\bf k}) + z{'} ( n_1 {\bf i} + n_2 {\bf j} + n_3 {\bf k}) \\ &=& (\ell_1 x{'} + m_1 y{'} +n_1 z{'}){\bf i} + (\ell_2 x{'} + m_2 y{'} +n_2 z{'}){\bf j} + (\ell_3 x{'} + m_3 y{'} +n_3 z{'}){\bf k} \end{eqnarray} Comparing the last expression with \({\bf r}= x{\bf i} + y {\bf j} + z{\bf k}\), we get \begin{equation} x= (\ell_1 x{'} + m_1 y{'} +n_1 z{'});\quad y= ( m_1 {\bf i} + m_2 {\bf j} + m_3 {\bf k});\quad z = (\ell_3 x{'} + m_3 y{'} +n_3 z{'}) \end{equation} Thus we get \begin{equation}\label{EQ12} \begin{pmatrix} x\\ y \\ z\end{pmatrix} = \begin{pmatrix} \ell_1 & m_1 & n_1\\ \ell_2 & m_2 & n_2\\ \ell_3 & m_3 & n_3\\ \end{pmatrix} \begin{pmatrix} x{'} \\ y{'} \\ z{'} \end{pmatrix} \end{equation}

Get the rotation matrix

Using the fact that vectors \({\bf l}, {\bf m}, {\bf n}\) are pairwise orthogonal unit vectors, it is easy to see that the inverse relation is given by \begin{equation}\label{EQ13} \begin{pmatrix} x{'} \\ y{'} \\ z{'} \end{pmatrix} = \begin{pmatrix} \ell_1 & \ell_2 & \ell_3\\ m_1 & m_2 & m_3\\ n_1 & n_2 & n_3\\ \end{pmatrix} \begin{pmatrix} x\\ y \\ z\end{pmatrix} \end{equation} We introduce the notation \begin{equation}\label{EQ14} R = \begin{pmatrix} \ell_1 & \ell_2 & \ell_3\\ m_1 & m_2 & m_3\\ n_1 & n_2 & n_3\\ \end{pmatrix} , \end{equation} and also a column vector notation for vectors, \begin{equation} \widetilde{\sf r} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} \qquad \widetilde{\sf r}{'} = \begin{pmatrix} x{'} \\ y{'} \\ z{'} \end{pmatrix} . \end{equation} The transformation equation \eqRef{EQ13}, relating the components of {\bf r} w.r.t the coordinate frames \(K,K{'}\), takes a compact form \begin{equation} \label{EQ15} \widetilde{\sf r}{'} = R \,\widetilde{ \sf r} \end{equation} The matrix \(R\) will be called rotation matrix for change of axes reference frame from \(K,K{'}\). 

Important:

• When working with only one coordinate system there is no need to distinguish between \({\mathbf A}\) and \(\vec{A}\). These two can be used interchangeably.
• When working with two or more coordinate systems \(K{'}, K{''}, \ldots\), we use \(\vec{A}{'}, \vec{A}{''}\) to denotes components w.r.t systems \(K{'}, K{''}, \ldots\).
• The components w.r.t. different coordinate systems will be collectively written as \begin{eqnarray} \vec{A}{'}=(A_x{'},A_y{'},A_z{'}); \quad \text{and}\quad \vec{A}{''}=(A_x{''},A_y{''},A_z{''}) \end{eqnarray}
• The vector itself can be written as \begin{eqnarray} {\mathbf A} &=& A{'}_x \hat{i}{'} + A{'}_y \hat{j} + A{'}_z \hat{k}\\ &=&A_x{''} \,\hat{i}{''} + A_y{''}\, \hat{j}{''} + A_z{''}\,\hat{k}{''} \end{eqnarray}
• Frequently, following matrix notation of assembling the components of a vector in a column vector turns out to be very convenient. A vector \(\mathbf A\) in the matrix notation will be denoted by \(\widetilde{\sf A}\), where \begin{equation} \widetilde{\sf A} = \begin{pmatrix} A_x \\ A_y \\ A_z \end{pmatrix} \end{equation}
• Finally, the "\(1-2-3\)" notation for the components of a vector \(\vec{A}=(A_1, A_2, A_3)\) will also be used in place of "{\tt x-y-z}" notation \(\vec{A}=(A_x,A_y,A_z)\).Questions for you

1. Verify that the transpose of matrix \(R\) equals the inverse of \(R\), \(R^TR=I\) and that \(\det R=1\).}
2. Using orthogonality property of the rotation matrix, show that the dot product of two vectors remains same when computed using the components in two different coordinate systems. This is expected as the values of length of a vector and angles between two vectors does not depend upon the choice of coordinate system.