I want you to think of an arrow in another way... Here is an arrow... Now if we multiply, you have to think in a different way than for adding. There… - Richard Feynman

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I want you to think of an arrow in another way... Here is an arrow... Now if we multiply, you have to think in a different way than for adding. There's an arrow... and imagine there's a [different] standard arrow... always horizontal and has unit length, that's the standard unit arrow. Now suppose I have a second arrow and I want to multiply them... [W]hat do I mean by multiplying? ...Let me first describe this [first] arrow [number 1] ...compare it to the standard arrow and ask for the relation... You can turn... and shrink it. So an arrow describes... how much I have to shrink the standard, and how much I have to rotate it to get the arrow I want. Now multiplication of arrows means that you do these rotations and shrinkings in succession. ...Now if I take this arrow [#2] ...this red [arrow #3] is the product [of arrow #1 and arrow #2].... It bears the same geometric relationship to the purple arrow [#2] as the blue one [arrow number 1] bears to the black one [standard arrow]. In other words it's supposed to be turned the same degree and shrunk the same degree as the blue one [arrow #2] is to the black [standard] one. In other words this [arrow #1] is to that [standard arrow], as this [arrow #3] arrow is to that [arrow #2].