School of Mathematics and Statistics, The University of Sydney
 11. The vector product
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The algebraic rules of the vector product

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The vector product obeys the following algebraic rules

1. u × v = - (v × u);  2. s(u ×  v) = (sv) × u = v ×  (su)  for every scalar s;  3. (u +  v) × w =  u × w + v ×  w;

Warning

The expression (u × v) × w makes sense, but

(u ×  v) × w /=  u × (v × w)

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 = u1i + u2j + u3k, v = v1i + v2j + v3k and w = w1i + w2j + w3k.

  1. u×v = -(v×u) follows directly from the definition of the vector product;
  2. For every scalar s
    s(u ×  v) = s(u2v3-  u3v2)i + s(u3v1 - u1v3)j + s(u1v2- u2v1)k         = (su  v - su  v )i + (su v  - su v )j + (su v - su  v )k               2 3     3 2       3 1     1 3        1 2     2 1                                                  = (sv) × u = v ×  (su)
  3. Using the Cartesian representations we get
    (u + v) × w  = ((u  + v )w  - (u  + v )w )i                   2    2   3    3    3  2      +  ((u3 + v3)w1 -  (u1 + v1)w3)j + ((u1 + v1)w2 - (u2 + v2)w1)k           = (u2w3 - u3w2)i + (u3w1  - u1w3)j + (u1w2 - u2w1)k            +  (v2w3 -  v3w2)i + (v3w1 -  v1w3)j + (v1w2 -  v2w1)k                                                      = u × w  + v × w.

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