Conceptual Understanding in Introductory Physics XXIX: Two Expressions for the Dot Product

Students sometimes see vector dot products in their calculus classes before they see them in their physics classes. Dot products are often presented with two seemingly unrelated definitions, one of which is geometric and coordinate free and the other is in terms of components in a particular basis. Yet, the two give exactly the same number, which is rather counterintuitive.

Given two vectors \mathbf{a} and \mathbf{b}, their dot product is given by \mathbf{a}\bullet\mathbf{b} = \left\lVert\mathbf{a}\right\rVert \left\lVert\mathbf{b}\right\rVert \cos\theta, which in words is just the product of the vectors’ magnitudes and the cosine of the angle between them. But students also learn that the dot product can be calculated in terms of the scalar components as \mathbf{a}\bullet\mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3, which in words is just the sum of products of corresponding scalar components of the two vectors.

Here’s the question: Show that these two expressions for \mathbf{a}\bullet\mathbf{b} are equal; show that \left\lVert\mathbf{a}\right\rVert \left\lVert\mathbf{b}\right\rVert \cos\theta = a_1 b_1 + a_2 b_2 + a_3 b_3. (HINT: Make a drawing and use the law of cosines.)

I think this question is extremely important because it connects two concepts that at first glance look totally unrelated. No one ever showed me this as a student, and indeed I doubt very seriously that most introductory physics students have seen it.


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