For the following two straight lines it shall be proved: are they parallel, do they have a point in common, or are these properties not valid. |
g : x = a_{1} + m * b_{1} h : x = a_{2} + l * b_{2}
a) First I try to prove if the
straight lines are parallel.
To do this I consider the vectors which define the direction of the straight lines. These are the vectors shown in the picture above which have been multiplied with m resp. l. If the two straight lines are parallel then a real
number exists, multiplied with the
first direction vector b_{1},
results in the second direction vector b_{2}.
It will be shown that this assumption leads to a
contradiction. Note: the symbols ^{.} and * denote the
scalar product of vectors. |
Translation:
Annahme = assumption
und = and
das ist ein Widerspruch = this is
a contradiction
b) Now I try to prove if a common point
of the two straigth lines exists. If it exists, the
coordinates of this common point will be calculated.
Consider the following equation (looking for a common point): |
Translation:
ist ein Widerspruch = this is a contradiction
Die beiden Geraden haben keinen gemeinsamen Punkt = the
two straight lines have no point in common
It follows that the two straight lines are not parallel and that they have no point in common. In German: they are "windschief" (English "cockeyed" ?). |
The foregoing picture shows how the
shortest distance between the two straight lines can be
calculated.
The vector with endpoint P_{1} is calculated as a_{1}
+ l·b_{1} b_{1} and b_{2} are vectors in the corresponding directions of the straight lines._{} The vector which connects P_{1} and P_{2}
is calculated to it has the shortest length that is possible if it is perpenticular to the straight lines that is ((a_{1} + l·b_{1})
- (a_{2} + m·b_{2}))
° b_{1} = 0 Now l and m can be calculated The distance d between the two straight lines is
calculated as In the following text l ist replaced by and m is replaced by |