Scalars & Vectors in Terms of
Transformations: Sect. 1.8
• Skip Sects. 1,5, 1.6, 1.7 for now. Possibly come back
to later (as needed!)
• Consider the orthogonal coordinate transformation of
the type:
xi = ∑j λij xj
(i,j = 1,2,3)
(1)
with ∑j λij λkj = δik ,
(2)
Definition of a Scalar
• If, under the orthogonal transformation
defined by (1) & (2), a quantity  is
unaffected:   Scalar
• Consider again an orthogonal coordinate transformation
of the type: xi = ∑j λij xj (i,j = 1,2,3)
(1)
with
∑j λij λkj = δik ,
(2)
Definition of a Vector
• Consider three quantities (A1,A2,A3)
• If, under the orthogonal transformation defined
by (1), (2), (A1,A2,A3) are changed to (A1, A2,A3)
& the relation between the primed primed &
unprimed quantities satisfies (1): [The same ij as in
Ai = ∑j λij Aj
(i,j = 1,2,3)
A = (A1,A2,A3)  Vector
(1)!]
Elementary Scalar & Vector Operations
Sect. 1.9: No Proofs, only results!
• Consider 3 Vectors: A = (A1,A2,A3), B = (B1,B2,B3),
C = (C1,C2,C3). 3 Scalars: φ,ψ,ξ
• Elementary scalar & vector algebra:
Commutative Law
Ai+Bi = Bi+Ai ;
φ+ψ=ψ+φ
Associative Law
Ai+(Bi +Ci) = (Ai+Bi)+ Ci ; φ + (ψ + ξ) = (ψ + φ) + ξ
Multiplication of a vector by a scalar
ξφ = ψ (a scalar!);
ξA = B
(a vector!)
Scalar Product of 2 Vectors: Sect. 1.10
• Consider 2 Vectors:
A = (A1,A2,A3), B = (B1,B2,B3).
• Definition of the scalar (dot) product:
AB  ∑i Ai Bi
(1)
– The magnitude (length) of A:
A= |A|  [(A1)2+(A2)2+(A3)2]½
• Divide both sides of (1) by AB:
(AB)/(AB) = ∑i (Ai Bi)/(AB)
(AB)/(AB) = ∑i (Ai Bi)/(AB)
• Suppose A makes angle α with
the x1 axis (Fig.):  A1/A = cosα
(a direction cosine of A).
 Can write in general:
(AB)/(AB) = ∑i (ΛA)i(Λ B)i
where:
(ΛA)i  Ai/A; (ΛB)i  Bi/B 
direction cosines of A & B
By an earlier identity we can write:
∑i (ΛA)i(ΛB)i = cos(A,B)

AB = AB cos(A,B)
• See text for proof that AB is a scalar & obeys
commutative & distributive laws.
• Special cases: Consider distance from origin to
(x1,x2,x3) = Magnitude (length) of vector r:
r = |r|  [rr]½  [(x1)2+(x2)2+(x3)2]½
Likewise, distance from origin to (x1,x2,x3) =
length of r:
r = |r|  [rr]½  [(x1)2+(x2)2+(x3)2]½
Similarly, the distance from (x1,x2,x3) to (x1,x2, x3) =
length of r - r:
d=|r-r|=[(r-r)(r-r)]½  [(x1-x1)2+(x2-x2)2+(x3-x3)2]½
• Bottom line: The distance between
2 points in 3d space is the square
root of a scalar product.  The
distance between 2 points is
invariant under orthogonal
coordinate transformations!
Unit Vectors: Work Example 1.5!
• To describe vectors in terms of components along
various axes, it is useful to use unit vectors.
• Unit vectors: Magnitude = 1 along specified axes.
• Example, unit vector in R direction is eR = R/|R|
• Various symbols, relevant to various coordinate
systems: (i,j,k), (e1,e2,e3), (er,eθ,e), (r,θ,), ..
• Can write:
A= (A1,A2,A3) = A1e1+ A2e2+A3e3= ∑i Aiei
= A1i+ A2j+A3k Also: Ai= Aei
If unit the vectors are orthogonal, as they usually are, then, we must have
eiej = δij. From now on, unless there might be some confusion, a vector
will be a bold letter (A) & the “hat” will be left off of the unit vectors.
Vector Product of 2 Vectors
• The Vector (or cross) Product of 2 vectors: A
VECTOR. Write C = A  B
– Cartesian components of C: Ci  ∑j,k εijk AjBk
εijk  permutation symbol or Levi-Civita density
εijk  0, if any 2 indices equal
 1, if i, j, k form even permutation of 1,2,3
 -1, if i, j, k form odd permutation of 1,2,3
ε122 = ε313 = ε211 = 0, etc.; ε123= ε231= ε312= 1, ε132= ε213= ε321= -1
 C1 = A2B3 - A3B2 , C2 =A3B1 - A1B3
C3 = A1B2 - A2B1
• C =A B
– The text proves that:
|C| =|A||B|sin(A,B)
• Also, C is  to the plane formed by A & B
(figure):
• Work Example 1.6!
• Properties of the vector product:
A B=-BA
(A  B)  C  A  (B  C)
(A  B)  C = B(AC) - C(AB)
• Unit vector orthogonality
 ei  ej = ek i,j,k in cyclic order
• Can also write ei  ej = ∑k εijk ek
• Can show that (Cartesian coordinates only!):
e1 e2 e3
C=A B=
A1 A2 A 3
(determinant)
B1 B2 B3
• Various vector identities:
• Work Example 1.7!!