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Why was $\vec{A}\cdot\vec{B}$ defined as $|\vec{A}||\vec{B}|\cos \theta$? Historically what is the underlying idea?

kiss my armpit
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    It gives the length of the projection of $A$ onto $B$, weighted by $|B$|, right? (Or vice versa.) – Potato Aug 29 '13 at 02:30
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    You can also look at it as an inner product and the angle $\theta$ appears as a balancing factor vis-a-vis the Cauchy-Schwarz Inequality. It serves as a motivation for defining an angle in some respects. – Vishesh Aug 29 '13 at 02:34
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    Yes. It's intuitively a good way so show how close or how far away to orthogonality we are if we are solving for $\theta$. – Eleven-Eleven Aug 29 '13 at 02:35
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    I've never seen it defined that way, but as a consequence of geometric reasoning. – Cameron Williams Aug 29 '13 at 02:38

2 Answers2

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I think it's the other way round. First, people became interested in the projection of one vector onto another, $|a||b|\cos(\theta)$, then saw this was equivalent to the usual inner product in $\mathbb{R}^n$ $(\sum x_i y_i)$, and only afterwards was a general notion of an inner product space conceived, as a generalization of the intuitive space in $\mathbb{R}^3$.

For more on the topic, I'd suggest A History of Vector Analysis , in which the author expounds upon Grassman's initial discoveries and definition of the inner product, as well as Gibbs's later independent dicoveries and usage.

Nathaniel Bubis
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    It indeed historically was the other way: using vectors to express certain geometrical concepts came after the development of the trigonometric formulas. The idea of using vector products to notate the operations we associate with them is attributed to Oliver Heaviside and Josiah Willard Gibbs, so the "dot" and "cross products" only go back about 120-130 years. (Maxwell, for instance, wrote out his eponymous equations in component form...) – colormegone Aug 29 '13 at 02:56
  • @RecklessReckoner - it actually looks like Grassman invented the inner product (along with most of Linear Algebra) in 1832, though it looks like not many people appreciated it at the time. – Nathaniel Bubis Aug 29 '13 at 02:59
  • Ah well, in mathematics, as in so many other human activities, the "price of being the pioneer" is that one's work often goes unappreciated for decades or generations before enough other people realize the value of the ideas (and then frequently attribute them to someone contemporary)... – colormegone Aug 29 '13 at 03:13
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Because it is fundamentally linked to the solution of practical problems in mathematics and physics. In physics we often find ourselves needing to decompose a vector into components. For example when we think about acceleration as a vector, we are often concerned with its components in the tangential and normal directions. The dot product helps us do that. The dot product also shows up in the definition of work and flux. In mathematics it is prevalent in linear algebra as well as vector algebra and vector calculus (Divergence Theorem, Stokes Theorem). Because the idea of projection appears in so many places, it warranted its own definition and notation.

Tpofofn
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