2. 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: x i = ∑ j λ ij x j (i,j = 1,2,3) (1) with ∑ j λ ij λ kj = δ ik, (2) Definition of a Scalar If, under the orthogonal transformation defined by (1) & (2), a quantity  is unaffected:   Scalar

3. Consider again an orthogonal coordinate transformation of the type: x i = ∑ j λ ij x j (i,j = 1,2,3) (1) with ∑ j λ ij λ kj = δ ik, (2) Definition of a Vector Consider three quantities (A 1,A 2,A 3 ) If, under the orthogonal transformation defined by (1), (2), (A 1,A 2,A 3 ) are changed to (A 1, A 2,A 3 ) & the relation between the primed primed & unprimed quantities satisfies (1): A i = ∑ j λ ij A j (i,j = 1,2,3) A = (A 1,A 2,A 3 )  Vector [The same  ij as in (1)!]

4. Elementary Scalar & Vector Operations Sect. 1.9: No Proofs, only results! Consider 3 Vectors: A = (A 1,A 2,A 3 ), B = (B 1,B 2,B 3 ), C = (C 1,C 2,C 3 ). 3 Scalars: φ,ψ,ξ Elementary scalar & vector algebra: Commutative Law A i +B i = B i +A i ; φ + ψ = ψ + φ Associative Law A i +(B i +C i ) = (A i +B i )+ C i ; φ + (ψ + ξ) = (ψ + φ) + ξ Multiplication of a vector by a scalar ξφ = ψ (a scalar!); ξA = B (a vector!)

5. Scalar Product of 2 Vectors: Sect. 1.10 Consider 2 Vectors: A = (A 1,A 2,A 3 ), B = (B 1,B 2,B 3 ). Definition of the scalar (dot) product: A  B  ∑ i A i B i (1) –The magnitude (length) of A: A= |A|  [(A 1 ) 2 +(A 2 ) 2 +(A 3 ) 2 ] ½ Divide both sides of (1) by AB: (A  B)/(AB) = ∑ i (A i B i )/(AB)

6. Suppose A makes angle α with the x 1 axis (Fig.):  A 1 /A = cosα (a direction cosine of A).  Can write in general: (A  B)/(AB) = ∑ i (Λ A ) i (Λ B ) i where: (Λ A ) i  A i /A; (Λ B ) i  B i /B  direction cosines of A & B By an earlier identity we can write: ∑ i (Λ A ) i (Λ B ) i = cos(A,B)

7.  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 (x 1,x 2,x 3 ) = Magnitude (length) of vector r: r = |r|  [r  r] ½  [(x 1 ) 2 +(x 2 ) 2 +(x 3 ) 2 ] ½ Likewise, distance from origin to (x 1,x 2,x 3 ) = length of r: r = |r|  [r  r] ½  [(x 1 ) 2 +(x 2 ) 2 +(x 3 ) 2 ] ½ Similarly, the distance from (x 1,x 2,x 3 ) to (x 1,x 2, x 3 ) = length of r - r: d=|r-r|=[(r-r)  (r-r)] ½  [(x 1 -x 1 ) 2 +(x 2 -x 2 ) 2 +(x 3 -x 3 ) 2 ] ½

8. 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!

9. 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 e R = R/|R| Various symbols, relevant to various coordinate systems: (i,j,k), (e 1,e 2,e 3 ), (e r,e θ,e  ), (r,θ,  ),.. Can write: A= (A 1,A 2,A 3 ) = A 1 e 1 + A 2 e 2 +A 3 e 3 = ∑ i A i e i = A 1 i+ A 2 j+A 3 k Also: A i = A  e i If unit the vectors are orthogonal, as they usually are, then, we must have e i  e j = δ ij. 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.

10. Vector Product of 2 Vectors The Vector (or cross) Product of 2 vectors: A VECTOR. Write C = A  B –Cartesian components of C: C i  ∑ j,k ε ijk A j B k ε ijk  permutation symbol or Levi-Civita density ε ijk  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 ε 122 = ε 313 = ε 211 = 0, etc.; ε 123 = ε 231 = ε 312 = 1, ε 132 = ε 213 = ε 321 = -1  C 1 = A 2 B 3 - A 3 B 2, C 2 =A 3 B 1 - A 1 B 3 C 3 = A 1 B 2 - A 2 B 1

11. 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!

12. 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  e i  e j = e k i,j,k in cyclic order Can also write e i  e j = ∑ k ε ijk e k Can show that (Cartesian coordinates only!): e 1 e 2 e 3 C = A  B = A 1 A 2 A 3  (determinant) B 1 B 2 B 3

13. Various vector identities: Work Example 1.7!!