Problem 1:
Let
be the point with coordinates
with respect to
the Cartesian coordinate system spanned by the canonical unit vectors of
.
Find the spherical and the cylindrical coordinates of
. Which Cartesian coordinates does
point
have, if we rotate the coordinate system about the
-axis by
.

Solution:
Spherical coordinates of
:
,
,
Cylindrical coordinates of
:
,
,
Cartesian coordinates after rotation (Results should be rounded to 4 decimal digits):
,,

Problem 2:
Let
be vectors in
.
Decide whether the following statements are true or false.

a)

implies that
at least one of the two vectors
is the zero vector.

b)

The following holds:
.

c)

Every orthonormal basis form a right-handed system.

d)

The following holds:
.

e)

If vector
is a multiple of vector
, then
.

Solution:

a)

true

false

b)

true

false

c)

true

false

d)

true

false

e)

true

false

Problem 3:
Show that the vectors

are mutually orthogonal. Do they form a left-handed or a right-handed system?

Keine Angabe , left-handed ,
right-handed .

Calculate the magnitudes , , and find
so that

Solution:

Magnitudes:
,
,
.

Parameter:
, , .

Problem 4:
Given the points
,
and
in
. Let
be the line
through
and
, and let
be the line through
with direction

Find the distance of
from
, and the distance between
and
.

Solution:

Distance of
from
: .

Distance between
and
: .

Problem 5:
Given the following plane in

a)

Let
be the plane through point
, parallel
to
. Find the equation describing the plane
.

b)

Which point
on plane
has minimal
distance from point
. What is the minimal distance?

c)

Show that point
lies in the plane
, and that
the points
form an equilateral triangle.
Find the lenghts of the sides, all interior angles, and the area of the triangle.

Solution:

a)

Complete the missing coefficients of the equation of
:
.

b)

Point
, ,
, distance: .

c)

squared sides:
,
,
.
,
,
.
Area of triangle: /
(given as completely reduced fraction).