Brandenburg Technical University (BTU) Cottbus
Chair of Mathematics for Enineering Prof. Dr. R. Reemtsen, Dr. F. Kemm
Mathematics for Engineering I
Problem Sheet No. 7, December 03/04, 2007 www.math.tu-cottbus.de/˜kemm/lehre/erm
Class Problems
1. Check whether the lines g1, g2 are skew, parallel, or intersect in one point. If they intersect, also try to determine the point of intersection. If they are skew lines, determine the distance between g1 and g2.
a)
g1 : ~x=
1 0 0
+t
−1 1 1
; g2 : ~x=t
1 1 1
.
b)
g1 : ~x=t
2
−1 2
; g2 : ~x=
2 2 0
+t
0 0 1
.
2. Let E be a plane in space, defined by x1+x2−x3 =−4.
a) Rewrite the plane in the Hesse normal form.
b) Find a parameter representation of this plane.
b) Find a lineg that is orthogonal to E.
3. Let a lineg1be given by the equationsx1+x2 = 1 andx3 = 0. Find the intersecting line which is orthogonal to g1 and runs through the point (1,1,1).
4. For the line in the plane given by the pointsA= (2,3) andB = (−10,−2) compute a) a parameter representation of the line,
b) the Hesse normal form of the line, c) the distance of the line to the origin,
d) the shortest position vector pointing from the origin to the line.
Homework
1. Check whether the lines g1, g2 are skew, parallel, or intersect in one point. If they intersect, also try to determine the point of intersection. If they are skew lines, determine the distance between g1 and g2.
a)
g1 : ~x=
0 1 2
+t
3 2 2
; g2 : ~x=
1 1 1
+t
4 2 1
.
b)
g1 : ~x=t
2 3 1
; g2 : ~x=
1 2 3
+t
3 1 2
.
2. Let E be a plane in space, defined by 2x1−x2−3x3 = 2.
a) Rewrite the plane in the Hesse normal form.
b) Find a parameter representation of this plane.
c) Find a lineg that is orthogonal to E.
3. The line g1 runs through the points (1,−1,1) and (−1,−1,−1).
The line g2 runs through (−1,1,1) and (1,1,−1).
a) Give a point-direction equation for each of both lines.
b) Evaluate the points of intersection of the lines gi with the plane x3 = 3.
4. Let a lineg1be given by the equationsx2+x3 = 1 andx1 = 0. Find the intersecting line which is orthogonal to g1 and runs through the point (2,−1,3).
5. For the plane in the three dimensional space which is given by the points A= (1,0,0), B = (3,−2,2) andC = (2,−4,−2) compute
a) a parameter form of the plane, b) the Hesse normal form of the plane,
c) the distance of the plane to the origin,
d) the shortest position vector pointing from the origin to the plane.
6. Solve the following equations and inequalities:
|x−5| ≤ |x2−7|,
|x+ 2| − |2−x|= 2|x|. 7. Solve the following equations in the complex plane:
z9 =−1, z2+ 2i= 1.
