11. The vector product





The vector product obeys the following algebraic rules
WarningThe expression (u × v) × w makes sense, but in general. As a counterexample look at (i × i) × j. We have (i × i) × j = 0 × j = 0 but i × (i × j) = i × k = j. This shows that writing u × v × w does not make sense as it is not clear which two vectors to multiply first! The easiest way to prove the algebraic rules of the vector product is to make use of the Cartesian representation of the vector product. Hence assume that u = u_{1}i + u_{2}j + u_{3}k, v = v_{1}i + v_{2}j + v_{3}k and w = w_{1}i + w_{2}j + w_{3}k.


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