We have a line in <mi mathvariant="double-struck">R 3 </msup> given as interse

madridomot

madridomot

Answered question

2022-05-25

We have a line in R 3 given as intersetion of two planes:
{ A 1 x + B 1 y + C 1 z + D 1 = 0 A 2 x + B 2 y + C 2 z + D 2 = 0
How to represent it in parametric form:
{ x = x 0 + a t y = y 0 + b t z = z 0 + c t ?

Answer & Explanation

bideanbarrenaf5

bideanbarrenaf5

Beginner2022-05-26Added 8 answers

What you look for is a point on the line ( x 0 , y 0 , z 0 ) , and a direction vector of the line ( a , b , c ) .
To find a point on the line, you can for example fix x and find y , z (there are some case where this won't work).
To find a direction vector, note that the vector ( A 1 , B 1 , C 1 ) is orthogonal to the first plane therefore to the line. Likewise, ( A 2 , B 2 , C 2 ) is orthogonal to the line. If you take their vector product you will get a direction vector.
Another way to find a direction vector, is to find another point on the line and subtract one point from the other.
Liberty Mack

Liberty Mack

Beginner2022-05-27Added 6 answers

Row reduce the system of equations. Generically, you will get an equation with two variables and one with three. There is going to be a repeated variable in both, that is your parameter. Write the remaining variables in terms of this common variable and you are done.
In your example:
l : { x + y z + 1 = 0 x y + z 1 = 0
Add the first equation to the last to get:
l : { x + y z + 1 = 0 x = 0
We can take y or z as the parameter. For example let's take z. Then
x = 0 + 0 z y = 1 + z z = 0 + z

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