Appendix
Exercises: Answers and Solutions
Exercise Chapter 1
1.1 Complex Numbers as 2D Vectors(p.6)
Convince yourself that the complex numbers z=x+i y are elements of a vector
space, i.e. that they obey the rules(1.1)–(1.6).Make a sketch to demonstrate that
z 1 +z 2 =z 2 +z 1 ,with z 1 = 3 + 4 i and z 2 = 4 + 3 i,in accord with the vector
addition in 2 D.
Exercise Chapter 2
2.1 Exercise: Compute Scalar Product for given Vectors(p.14)
Compute the length, the scalar products and the angles betwen the three vectorsa,
b,cwhich have the components{ 1 , 0 , 0 },{ 1 , 1 , 0 },and{ 1 , 1 , 1 }.
Hint:to visualize the directions of the vectors, make a sketch of a cube and draw
them there!
The scalar products of the vectors with themselves are:a·a=1,b·b=2,
c·c=3, and consequently
a=|a|= 1 , b=|b|=√
2 , c=|c|=√
3.
The mutual scalar products are
a·b= 1 , a·c= 1 , b·c= 2.The cosine of the angleφbetween these vectors are
1
√
2,
1
√
3
,
2
√
6
,
© Springer International Publishing Switzerland 2015
S. Hess,Tensors for Physics, Undergraduate Lecture Notes in Physics,
DOI 10.1007/978-3-319-12787-3
389