Invariance of Physical Laws - Vectors and Vector Operations

  1. Introduction

    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.

  2. Objectives

    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.

  3. Materials

    Notebooks and paper are required. Some students may want to use calculators but they are not needed.

  4. Vocabulary
  5. Presentation

    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
    Cartesian unit vectors cylinderical unit vectors spherical unit vectors

    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°.

    Unit vector properties.

    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
    cylindrical unit vectors spherical unit vectors

    Kinematics expressions that show independence of motion in three orthogonal directions.

    Cartesian unit vectors cylindrical unit vectors

    Combining the three independent kinematic expressions into one vector expression.

    Cartesian unit vectors cylindrical unit vectors

    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 definition of vector 1
    Vector 2 definition of vector 2
    Vector addition vector addition
    Scalar product scalar produce
    Vector product vector product
  6. Evaluation

    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.