2.1 Coordinate System and Position Vector 13
The vectorr, divided by its lengthr, is the dimensionlessunit vector̂r:̂r=r−^1 r, (2.5)or, in component notation:
̂rμ=r−^1 rμ. (2.6)
The unit vector has magnitude 1:
̂rμ̂rμ= 1. (2.7)2.1.3 Scalar Product
The scalar product of two position vectorsrandswith componentsrμandsμis
r·s=r 1 s 1 +r 2 s 2 +r 3 s 3 :=rμsμ. (2.8)Clearly, the length squared (2.3) of the vectorris its scalar product with itself. Just
as in (2.3), thecenter dot“·” is essential to indicate the scalar product, when the
vectors are written with bold face symbols. The summation convention is used for
the component notation. Notice that the “name” of the summation index does not
matter, i.e.rμsμ=rνsν=rλsλ. What really matters is: a Greek letter occurs twice
(and only twice) in a product.
The scalar product has a simple geometric interpretation. In general, the two
vectorsrandsspan a plane. We choose the coordinate system such thatris parallel
to the 1-axis andsis in the 1–2-plane, see Fig.2.2. Then the components ofrare
(r 1 , 0 , 0 )and those ofsare(s 1 ,s 2 , 0 ). The scalar product yieldsr·s=r 1 s 1 .The
lengths of the two vectors are given byr=r 1 ands=
√
s^21 +s^22. The angle between
randsis denoted byφ, see Fig.2.2. One hass 1 =rcosφ, and
r·s=rscosφ (2.9)Fig. 2.2 For the geometric
interpretation of the scalar
product