In this lesson we continue our presentation of kinematics by expanding how we use vectors, and vector operations. We will also examine orthogonal coordinate systems and transformations between coordinate systems.
We hope the student will understand the details of vector operations and how to transform between scalar expressions and vector expressions. The student should also understand that various coordinate systems can make analysis of problem easier or harder but the underlying physics is independent of the coordinate system.
Notebooks and paper are required. Some students may want to use calculators but they are not needed.
This lesson tends to be primarily a lecture format. Make every effort to get the students involved. It also covers a considerable amount of material. Spread it over two sessions if required.
The instructor should present vector representation of position, velocity, and acceleration. The rotational equivalents of these quantities should also be presented. However, be very cautious because the full rotational representation is probably beyond the scope of this course. Also present the full vector representation of the basic kinematic equations in both linear and rotational motion.
A concept of fundamental importance is that motion is independent in each of three orthogonal directions. This fact allows us to separate motion into three independent parts or combine motion into a vector notion.
Discuss orthogonal coordinate systems. Also discuss transformations between polar and two dimensional Cartesian systems. Three dimensional transformations between three dimensional Cartesian and spherical are good theory but don't spend too much time on it. Emphasize that the laws of physics are invariant with respect to the orthogonal coordinate system used to describe them.
Unit vectors.
Cartesian | Cylindrical | Spherical |
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Unit vectors: "Handedness" The cross product of two unit vectors at any point specifies the direction of the third unit vector. "Orthogonality" The dot product of any two unit vectors at any point is always zero because they point in directions that differ by 90°.
Cartesian definitions for position, velocity, and acceleration in vector format.
Kinematic expressions in Cartesian vector format.
Transformations between coordinate systems. (A single two dimensional example below.)
From polar to Cartesian | From Cartesian to polar |
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Kinematics expressions that show independence of motion in three orthogonal directions.
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Combining the three independent kinematic expressions into one vector expression.
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Vector operations expressed as operations on vector components. These expressions are exactly equivalent to the geometric interpretations of the operations expressed in the last session.
Vector 1 | ![]() |
Vector 2 | ![]() |
Vector addition | ![]() |
Scalar product | ![]() |
Vector product | ![]() |
There are plenty of text books that have elementary problems concerning vectors, coordinate systems, and kinematics. Pick a few appropriate problems to evaluate the students.