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Instructions and materials
Representations of rotations
This laboratory introduces you to the application of complex numbers for various applications. Perhaps the most general and fundamental is to represent rotations in 2-D space using a complex number. We can do this by considering the complex plane. In the complex plane the real axis corresponds to the x-axis (abcissa) and the imaginary axis corresponds to the y-axis (ordinate). Using this representation a single expression can designate a point in the 2-D space. The point is defined by A expi phi, where phi is the rotation angle with respect to the x-axis. Since phi has that designation we can rotate the point around a circle of radius A simply by changing the value of phi. This is the basic way in which a complex number can represent a rotation.
What is the connection with matrices?
A rotation can also be defined by a matrix. We can rotate a vector by the angle phi using the following rotation matrix:
Thus, we can see a connection between complex numbers and matrices, which explains why both topics are discussed together in this computational lab.
Applications of the mathematical methods
Rotations
There are numerous applications of rotations. One can think of angular momentum (both orbital and spin) in terms of the angular projections of each of these quantities on a Cartesian coordinate system. NMR experiments can be explained using a rotation operator picture, which is based on the ideas that we have developed here. Of course, group theory and representations of symmetry in molecular structure also use rotational transformations.
Matrix solution of linear equations
We will use matrix solutions of linear equations in the example of solving for the concentration of more than one molecule (analyte) of unknown concentration in a mixture. The matrix approach to this is shown in the segment below as an illustration of how to use matrix commands in Excel.
An alternative to this approach is to use singular value decomposition as discussed in the SVD section of lecture 3.
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