Brandenburg Technical University (BTU) Cottbus
Chair of Mathematics for Enineering Prof. Dr. R. Reemtsen, Dr. F. Kemm
Mathematics for Engineering I
Problem Sheet No. 6, November 26/27, 2007 www.math.tu-cottbus.de/˜kemm/lehre/erm
Class Problems
1. Find the distance between the points (2,3,−1) and (1,4,2).
2. Let ~a 6= ~0 and ~b 6= ~0 be two given vectors. Find the conditions for which the following relations are valid:
a) ~a·~b >0, b) ~a·~b= 0, d) |~a·~b|=|~a| |~b|. 3. Evaluate
~a×(~b×~c), (~a×~b)·~c, |~a|, |~c|, ~a·~b, ~c·~a for
~a=
1
−2 1
, ~b=
1 0 1
, ~c=
1 1 1
.
4. Calculate the area of the triangle defined by the points
A= (1,1,1), B = (4,2,0), C = (2,−3,3), From the result calculate the altitudes of the triangle.
5. Find ways to prove that the diagonals of a parallelogram intersect in their mid- points.
6. Compute the volume and the surface area of the tetrahedron with the following vertices:
A= (0,0,0), B = (2,0,0), C = (0,3,0), D= (1,1,4). Calculate also the heights of the tetrahedron.
7. Compute the volume and the surface area of the parallelepiped spanned by the vectors
~a=
1 0
−1
, ~b=
2 3 4
, ~c=
0 1
−1
.
From the results derive the altitudes of the parallelepiped.
Homework 1. Evaluate
|~a|, |~b|, |~c|, ~a·~b, ~b·~c, ~a·~c, ~a×~c, ~b×~c and (~a×~c)·~b for the given vectors
~a=
1 2 1
, ~b=
1
−1 0
, ~c=
1 1 1
2. Calculate the area of the triangle defined by the points A, B, C.
a) A= (1,1,1), B = (2,4,−1), C = (3,0,4) b) A= (1,1,1), B = (2,0,3), C = (0,2,−1)
From the results calculate the altitudes of the triangles.
3. Find ways to prove that the space diagonals of a parallelepiped intersect in their midpoints.
4. Compute the volume and the surface area of the tetrahedron with the following vertices:
A= (0,−1,0), B = (1,2,0), C = (0,−1,1), D = (1,1,4). Calculate also the heights of the tetrahedron.
5. Compute the volume and the surface area of the parallelepiped spanned by the vectors
~a=
3 0 0
, ~b=
2 3 4
, ~c=
0
−1 2
.
From the results derive the altitudes of the parallelepiped.
6. Simplify as much as you can:
2 ln(x) + ln(8x)−3 ln(2x) ;
q√ x+√
y·√ x+y
px2−y2 ; xny2−n
a2n+1b3n · a3b5n+2 x−2−ny3n+2 .
7. Solve the following equations for x∈R:
|x+ 1| − |x−1|=|x|, bx−a
b + a
x = b
x− b−ax
a .
