Exercises Linear Algebra - Straight Lines


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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 = a1 + m * b1  h : x = a2 + l * b2

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 b1, results in the second direction vector b2. It will be shown that this assumption leads to a contradiction.

Note: the symbols . and * denote the scalar product of vectors.

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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):

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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 P1 is calculated as a1 + l·b1
The vector with endpoint P2 is calculated as  a2 + m·b2

b1 and b2 are vectors in the corresponding directions of the straight lines.

The vector which connects P1 and P2 is calculated to 
(a1 + l·b1) - (a2 + m·b2)  

it has the shortest length that is possible if it is perpenticular to the straight lines

that is ((a1 + l·b1) - (a2 + m·b2)) ° b1 = 0 
 ((a1 + l·b1) - (a2 + m·b2)) ° b2 = 0
° describes the scalar product of vectors

Now l and m can be calculated

The distance d between the two straight lines is calculated as
d = |(a1 + l·b1) - (a2 + m·b2)| 

In the following text l ist replaced by lambdaand m is replaced by mue

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