14 2Basics
holds true. Or in words: the scalar product of two vectors is equal to the product of
their lengths times the cosine of the angle between them. The scalar product ofs
with the unit vector̂ris equal toscosφ. The vectorscosφ̂r=(s·̂r)̂ris called the
the projection ofsonto the direction ofr.
The value of the scalar product reaches its maximum and (negative) minimum
when the vectors are parallel (φ=0) and anti-parallel (φ=π). The scalar product
vanishes for two vectors which are perpendicular to each other, i.e. forφ=π/2.
Such vectors are also referred to asorthogonalvectors.
2.1 Exercise: Compute Scalar Product for Given Vectors
Compute the length, the scalar products and the angles between the vectorsa,b,c
which have the components{ 1 , 0 , 0 },{ 1 , 1 , 0 }, and{ 1 , 1 , 1 }.
2.1.4 Spherical Polar Coordinates
As stated before, the position vector√ rhas a length, specified by its magnituder=
r·r, and a direction, determined by the unit vector̂r,cf.(2.5) and (2.6). These parts
of the vector are often referred to asradial partandangular part. Indeed, the unit
vector and thus the direction ofrcan be specified by the twopolar anglesθandφ.
Conventionally, a particular coordinate system is chosen, the Cartesian coordinates
{r 1 ,r 2 ,r 3 }are denoted by{x,y,z}which, in turn, are related to thespherical polar
coordinates r,θ,φby
x=rsinθcosφ, y=rsinθsinφ, z=rcosθ. (2.10)Notice, the three numbers forr,θ,φare not components of a vector.
The information given by the Cartesian components of a unit vector corresponds
to a point on the unit sphere, identified by the two angles, similar to positions on earth.
Notice, however, that the standard choice made for the angleθwould correspond to
associateθ=0 andθ= 180 ◦with the North Pole and the South Pole, respectively,
whereas the equator would be atθ= 90 ◦. For positions on earth, one starts counting
θfrom zero on the equator and has to distinguish between North and South, or plus
and minus. In any case, the angle spans an interval of 180◦,orjustπ, whereas that
ofφis 360◦,or2π.
2.2 Vector as Linear Combination of Basis Vectors
2.2.1 Orthogonal Basis
Examples of orthogonal vectors are the unit vectorse(i),i =1, 2, 3, which are
parallel to the axes 1, 2, 3 of the Cartesian coordinate system. These vectors have
the propertiese(^1 )·e(^1 )=1,e(^1 )·e(^2 )= 0 ,..., in more general terms,