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Mathematics-Online test:

# Test 1 Vector Calculus

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).

 (conceptual design by Joachim Wipper) automatically generated 8/11/2017