Lecture 2, Wednesday, 8/27/08: Section 10.3: The dot product

Topics

• The dot product: Algebraic definition (in terms of components), and geometric interpretation (in terms of magnitudes and angles)
• Properties of the dot product (Th. 4.3, p. 819)
• Computing angle between vectors via dot products
• Perpendicular/orthogonal vectors
• Testing orthogonality of vectors
• Component and projection of a vector a onto another vector b (comp_ba and proj_ba)
• Computation of work by a given force over a given distance

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Homework (not collected)

Notes

• Note on trig values: Some of the above HW problems require numerical calculation of cosines and inverse cosines. These calculations are the least important important aspect of the problems. If you have a calculator handy, you can do these computations; otherwise, don't worry. In quizzes and exams (where calculators are not permitted) the problems will either be such that the trig values can be computed without calculator (e.g., cos^-1(1) or cos^-1(1/2)), or you would be asked to leave the answer in "raw form", such as cos^-1(3/5).
  (You should, of course, know the standard trig values like cos(0), cos(pi/6), etc.)
• Notation for the various products: Since there exist different types of products involving vectors (scalar multiplication, dot product, cross product), it is important to distinguish these through the notation. For products of scalars or a scalar times a vector, you can use a standard-sized dot, or leave out the dot (e.g., ca). However, to denote a dot product you should always use an explict, clearly visible, dot. Never write a dot product without such an explict dot as multiplication sign. Similarly, for the cross product, use an "x" as the multiplication sign.
• Component and projection formulas: The projection formula is a bit complicated. However, you can get by memorizing the (much simpler) formula for the component, if you remember how the projection is related to the component, namely, the component multiplied by a unit vector in b direction (i.e., b/||b||).
  The order of the two vectors (a and b) in these formulas is crucial. An easy way to remember the correct order is to note that the vector appearing in the subscript of the notation (i.e., the b in comp_ba) and in the denominator of the component formula (think of it as the "bottom" vector) is the vector onto which the projection occurs.

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Last modified Sun 31 Aug 2008 05:38:20 PM CDT