9. The scalar product (page 4)

School of Mathematics and Statistics, The University of Sydney

9. The scalar product



	Glossary



	Examples

	Quiz


Algebraic rules	Page 4 of 6

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

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

If s is a scalar then

(su) · v	= su₁v₁ + su₂v₂ + su₃v₃
	= u₁sv₁ + u₂sv₂ + u₃sv₃
	= s(u₁v₁ + u₂v₂ + u₃v₃) = s(u · v)

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)

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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Algebraic rules