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Unformatted text preview: The Geometry of the Dot and Cross Products Tevian Dray Department of Mathematics Oregon State University Corvallis, OR 97331 tevian@math.oregonstate.edu Corinne A. Manogue Department of Physics Oregon State University Corvallis, OR 97331 corinne@physics.oregonstate.edu January 15, 2008 Abstract We argue for pedagogical reasons that the dot and cross products should be defined by their geometric properties, from which algebraic representations can be derived, rather than the other way around. 1 Introduction Most students first learn the algebraic formula for the dot and cross prod ucts in rectangular coordinates, and only then are shown their geometric interpretations. We believe this should be done in the other order. Students tend to remember best the first definition they use; this should not be an algebraic formula devoid of context. The geometric definition is coordinate independent, and therefore conveys invariant properties of these products, not just a formula for calculating them. Furthermore, it is easier to derive the algebraic formula from the geometric one than the other way around, as we demonstrate below. 1 θ vector v vector v · vector w  vector w  vector w Figure 1: The dot product is fundamentally a projection. 2 Dot Product The dot product is fundamentally a projection. As shown in Figure 1, the dot product of a vector with a unit vector is the projection of that vector in the direction given by the unit vector. This leads to the geometric formula vector v · vector w =  vector v  vector w  cos θ (1) for the dot product of any two vectors vector v and vector w . An immediate consequence of (1) is that the dot product of a vector with itself gives the square of the length, that is vector v · vector v =  vector v  2 (2) In particular, taking the “square” of any unit vector yields 1, for example ˆ ı · ˆ ı = 1 (3) where ˆ ı as usual denotes the unit vector in the x direction. 1 Furthermore, it follows immediately from the geometric definition that two vectors are orthogonal if and only if their dot product vanishes, that is vector v ⊥ vector w ⇐⇒ vector v · vector w = 0 (4) For instance, if ˆ denotes the unit vector in the y direction, then ˆ ı · ˆ = 0 (5) 1 We follow standard usage among scientists and engineers by putting hats on unit vectors. 2 θ vector B vector C vector A Figure 2: The Law of Cosines is just the definition of the dot product! The geometry of an orthonormal basis is fully captured by these properties; each basis vector is normalized, which is (3), and each pair of vectors is orthogonal, which is (5). The components of a vector vector v in an orthonormal basis are just the dot products of vector v with each basis vector. For instance, in two dimensions, setting v x = vector v · ˆ ı v y = vector v · ˆ (6) implies vector v = v x ˆ ı + v y ˆ . The component form of the dot product now follows from its properties given above. For example, if vector w = w x ˆ...
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This note was uploaded on 09/26/2011 for the course PHYSICS 106 taught by Professor Arubi during the Summer '11 term at UCLA.
 Summer '11
 Arubi
 Physics

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