School of Mathematics and Statistics, The University of Sydney
 9. The scalar product
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Algebraic rules

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Suppose u, v and w are vectors (in the plane or in space) with magnitudes |u|, |v| and |w|. Then

1. u · v = v · u;  2. s(u · v) = (su) · v = u(s · v) for every scalar s.  3. u · (v + w) = u · v + u · w;  4. u · u = |u|2.

The rules are obtained by using the Cartesian representation of the scalar product. We let u = u1i + u2j + u3k, v = v1i + v2j + v3k and w = w1i + w2j + w3k. Then

  1. By definition of the scalar product
    u · v = u1v1 + u2v2 + u3v3 = v1u1 + v2u2 + v3u3 = v · u

  2. If s is a scalar then
    (su) · v = su1v1 + su2v2 + su3v3
    = u1sv1 + u2sv2 + u3sv3
    = s(u1v1 + u2v2 + u3v3) = s(u · v)
  3. For the distributive law note that
    u · v + u · w
    = u1(v1 + w1) + u2(v2 + w2) + u3(v3 + w3)
    = u1v1 + u1w1 + u2v2 + u2w2 + u3v3 + u3w3
    = (u1v1 + u2v1 + u3v3) + (u1w1 + u2w2 + u3w3)
    = u · (v + w)
  4. For the last rule simply note that
    u · u = u1u1 + u2u2 + u3u3 = u21 + u22 + u23 = |u|2

Alternatively one could prove the algebraic rules using the Cartesian representation of the scalar product.

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