Parametric equations and lines

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SUMMARY

This discussion focuses on determining the relationships between four parametric lines: L1, L2, L3, and L4. The vectors associated with these lines are analyzed to establish whether they are parallel or identical. It is concluded that lines L1, L3, and L4 are parallel due to their direction vectors being scalar multiples of each other. The method for confirming if parallel lines are identical involves checking if a point from one line's equation satisfies the equation of another line.

PREREQUISITES
  • Understanding of parametric equations
  • Knowledge of vector operations and scalar multiplication
  • Familiarity with the concept of line equations in three-dimensional space
  • Ability to solve equations to find points of intersection
NEXT STEPS
  • Study the properties of vector equations and their geometric interpretations
  • Learn how to determine the intersection of parametric lines
  • Explore the concept of vector spaces and linear dependence
  • Practice problems involving parallel and identical lines in three dimensions
USEFUL FOR

This discussion is beneficial for students studying geometry, particularly those focusing on vector calculus and parametric equations, as well as educators teaching these concepts in mathematics courses.

Brigada
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Homework Statement


Determine if any of the lines are parallel or identical
L1 (x-8)/4 = (y+5)/-2 = (z+9)/3
L2 (x+7)/2 = (y-4)/1 = (z+6)/5
L3 (x+4)/-8 = (y-1)/4 = (z+18)/-6
L4 (x-2)/-2 = (y+3)/1 = (z-4)/1.5

Homework Equations


L1 pt(8,-5,-9) V<4,-2,3>
L2 pt(-7,4,-6) V<2,1,5>
L3 pt(-4,1,-18) V<-8,4,-6>
L4 pt(2,-3,4) V<-2,1,1.5>

The Attempt at a Solution


I know that if the vectors are scalar multiples, they are either parallel or identical. What I don't know, is after I find out that V(L3) = -2*V(L1). How do I determine if they are parallel or identical. I assume that since one vectors k value is 1.5, it is some multiple of another line, if not they wouldn't have given a 1.X. So L1 L3 L4 are parallel but how could I find if they were identical or not?
 
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After careful consideration, and pulling my head out of the book to think logically, I realized after I find the parallel lines, if a point I chose that satisfies one equation, also satisfies another line equation, they are identical, if its not on the line, it's still parallel, just not identical. That was 1.5 hours wasted on a brain fart!
 

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