[NOTES/CM-08014] Active and Passive Rotations

Active and passive rotations

In the  active point  of view of rotations the vectors are rotated. Under a rotation, a vector \(\vec x\) is mapped into another vector and we seek the relation between the two vectors.

In case of a passive rotation, the rotation is performed on the coordinate axes. In this case a given vector has different components in the two sets of axes. We will derive this relation between the components of position vector w.r.t two sets of axes.

In applications to physical theories these turn out to be just two different, but equivalent, ways of implementing of rotations. Both active and passive rotations are valid and can be used depending on the context and the mathematical formulation being employed.

A physical example of active rotation is the position vector of a point on a rotating rigid body. Two different observers may describe the motion of a body, or the same physical phenomenon, using two different coordinate systems related by a rotation. The motion of the body or set up, under observation, may or may not involve any rotation.

Relationship between active and passive views

It is of interest to note that transformation rule \begin{eqnarray}\Label{EQ01} \vec{x}^\prime = \vec{x} -(\hat{n}\times \vec{x}) \sin\alpha + \hat{n}\times(\hat{n}\times \vec{x})(1-\cos\alpha). \end{eqnarray} describes both active and passive rotations, when the symbols appearing in \eqRef{EQ01} are suitably interpreted as explained below. In \eqref{EQ01} we have used the notation
\begin{eqnarray} \vec x &=& (x_1,x_2,x_3) \\ \vec x^\prime &=&(x_1^\prime, x_2^\prime, x_3^\prime) \end{eqnarray}
The following points are to remembered when using \eqref{EQ01}. 

• The above quantities \(\vec x\equiv(x_1, x_2,x_3)\) and \(\vec x^\prime \equiv (x_1^\prime,x_2^\prime,x_3^\prime)\) and angle of rotation \(\alpha\) have different meanings in the two cases of the active and passive rotations.
• In the case of an active rotation, there is only one coordinate system. The vector \(\vec x^\prime\) is obtained by a clockwise rotation of the vector \(\vec x\) by an angle \(\alpha\).
• In the passive view the system of axes \(K^\prime\)is obtained by an {\tt anti clockwise rotation} of the coordinate axes \(K\). The quantities \(\vec x\) and \(\vec x^\prime\) are the components of the same vector \({\bf x}\) w.r.t. the two coordinate axes.
• Thus we see that he passive transformation of rotating the coordinate axes is equivalent to the active transformation of rotating the vector in the opposite direction in the sense described above.