Vectors and scalars – Introduction to the first classes in physics
for students of Macro (Faculty of Civil Engineering, MSc course, semester 1)
Marcin Miczek, PhD Eng in physics, Faculty o Mathematics and Physics, Institute of Physics,
Department of Applied Physics, building of Faculty of Electrical Engineering and of Faculty
of Mining and Geology (2 Krzywoustego), room no. 420. Classes: Thursday 10:45-11:30.
Consultation time: Monday 15:30-16:30, Thursday 16-17.
Problems for classes and other information will be published in our e-learning platform:
http://platforma.polsl.pl/rmf/
→ Instytut Fizyki, Wydział Budownictwa → Applied Physics
(semester 1) or shortly http://platforma.polsl.pl/rmf/course/view.php?id=196 Password is
applied-physics1
Evaluation of your work: classes (short tests and solving problems at the blackboard) – 40
points, final tests at the end of the semester (2 parts: theory and problems) – 60 points. Total:
maximum 100 points.
1.
Scalars and vectors – what are they, examples of physical quantities described by vectors
(displacement, velocity, acceleration, force, momentum) and scalars (mass, time, energy,
pressure, electric current).
2.
Notation:
a
a
AB
,
,
, a (in books).
3.
Features of a vector: magnitude/length/norm (conventional notation a=||a||), direction (in
Polish – „kierunek i zwrot”), initial point (essential for bound vectors and vector field) and
terminal point.
4.
Projection of a vector onto a line or onto a plane along a direction (direction of projection,
projection of a vector or a line segment parallel to the projection direction).
5.
Orthogonal projection. Cartesian coordinate systems 2D (OXY) and 3D (OXYZ).
Components of a vector. How to calculate components of a vector if the coordinates of its
initial and terminal point are known. Unit vectors (versors) i, j, k – usually with hat sign
above (^). Each vector can be expressed as a linear combination of versors:
[
]
k
a
j
a
i
a
a
a
a
a
z
y
x
z
y
x
ˆ
ˆ
ˆ
,
,
⋅
+
⋅
+
⋅
=
=
.
6.
Operations on vectors – geometrical constructions, analytical calculations, geometrical
interpretation:
a.
addition of vectors – triangle/polygon rule and parallelogram rule;
b.
opposite vector and subtraction of vectors;
c.
multiplication of a vector by a scalar
– summary: [a
x
, a
y
, a
z
] ± c·[b
x
, b
y
, b
z
] = [a
x
±c·b
x
, a
y
±c·b
y
, a
z
±c·b
z
], where c is a scalar;
d.
dot/inner/scalar product:
a
◦ b = a·b·cos(|
∠
(a,b)|)
[a
x
, a
y
, a
z
]
◦ [b
x
, b
y
, b
z
] = a
x
b
x
+a
y
b
y
+a
z
b
z
a
⊥
b ⇒ a
◦ b = 0 (scalar zero)
e.
cross/outer/vector product:
– descriptive definition: a
×
b=c that c
⊥
a, b (direction), c = a·b·sin(|
∠
(a,b)|)
(magnitude), (a, b, c) is right-handed system (direction);
a || b ⇒ a
×
b = 0 (null/zero vector)
– notation of vectors perpendicular to the plane of pictures (
or )
– determinant of a matrix:
[
] [
]
z
y
x
z
y
x
z
y
x
z
y
x
b
b
b
a
a
a
k
j
i
b
b
b
a
a
a
ˆ
ˆ
ˆ
,
,
,
,
=
×
