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

,

,

,

,

=

×

– geometrical interpretation: The magnitude of the cross product of vectors can be
interpreted as the area of the parallelogram having the vectors as sides.

See also: http://en.wikipedia.org/wiki/Vector_(spatial)