Solving Linear Intersections at 90°

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Homework Help Overview

The problem involves finding the vector, parametric, and symmetric equations of a line that intersects two given lines, L1 and L2, at right angles. The subject area pertains to vector geometry and line equations in three-dimensional space.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to use the cross product of direction vectors from the two lines to find a direction vector for the new line. Some participants suggest picking arbitrary points on each line to form a connecting line, while others inquire about the specific points needed for this approach.

Discussion Status

The discussion is ongoing, with participants exploring different methods to identify points on the lines and how to utilize the direction vector found. There is no explicit consensus yet, but guidance has been offered regarding the relationship between the points and the direction vector.

Contextual Notes

Participants are working under the constraints of the problem statement, which requires that the new line intersects both L1 and L2 at 90 degrees. There may be ambiguity regarding the choice of points on the lines and how to express the connecting line's equations.

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Homework Statement




Find the vector, parametric and symmetric equations of a line that intersect both line 1 and line 2 at 90°.

L1 :
x = 4 + 2t
y = 8 + 3t
z = -1 - 4t
L2 :
x = 7 - 6t
y = 2+ t
z = -1 + 2t

Homework Equations





The Attempt at a Solution



I tried taking the cross product of [2,3,-4], and [-6,1,2], to find the vector perpendicular to both lines


= [10,20,20]

So our new direction vector is [10,20,20]

Now I'm stuck (confused about how to find the equations)

Any help would be greatly appreciated!
 
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pick an arbitrary point on each line and write the equation of the line formed by joing the point, in terms of the points (use different parameteric variables for each line, say s & t)

then use the direction you have found to solve for s & t
 
Thanks lanedance,
could you elaborate a little more. what points would I need to pick, in order to solve for the equations.

Thanks,
 
you need to solve for the points that give you the connecting line, which is perpindicular to L1 & L2.

The first point, say p1, is a point on L1, the 2nd, p2 on L2. The direction of p1,p2 must be parallel or anti parallel to the direction you found. Use that fact to solve for p1 & p2 simultaneously.
 

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