Showing skew lines lie in parallel planes

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SUMMARY

This discussion focuses on determining the equations of two planes that contain skew lines L1 and L2. The directional vectors of the lines, <5, 5, -4> for L1 and <1, 8, -3> for L2, were used to compute the normal vector by crossing them. The participants clarified that to find the equation of a plane, one requires a normal vector and a point on the plane, which can be derived from the given lines. The need for more than one equation for a plane was also questioned, emphasizing that a single plane can be defined by its normal vector and a point.

PREREQUISITES
  • Understanding of vector operations, specifically cross products
  • Familiarity with the parametric equations of lines
  • Knowledge of the equation of a plane in 3D space
  • Basic concepts of skew lines in geometry
NEXT STEPS
  • Study the process of calculating the cross product of vectors
  • Learn how to derive the equation of a plane from a normal vector and a point
  • Explore the properties of skew lines and their geometric implications
  • Investigate the relationship between multiple equations of planes and their intersections
USEFUL FOR

Students and professionals in mathematics, particularly those studying geometry and vector calculus, as well as educators looking to explain the concepts of skew lines and planes.

mikemichiel
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Im givin these two lines..
L1= x=4+5t y=5+5t z=1-4t
L2= x=4+s y=-6+8s z=7-3s

What i tried doing was taking the directional vector of both lines <5 5 -4> <1 8 -3>, and crossing them to find the normal vector. I have enough information to find 1 equation of a plane, but how can I find the other. Can someone please point me in the right direction. Thanks in advance!
 
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Welcome to PF!

Hi mikemichiel! Welcome to PF! :wink:
mikemichiel said:
Im givin these two lines..
L1= x=4+5t y=5+5t z=1-4t
L2= x=4+s y=-6+8s z=7-3s

What i tried doing was taking the directional vector of both lines <5 5 -4> <1 8 -3>, and crossing them to find the normal vector. I have enough information to find 1 equation of a plane, but how can I find the other.

You have the normal …

can't you find the plane from that?

And why do you need more than 1 equation for a plane? :smile:
 
To find the equation of a plane, you need a normal vector and a point in the plane. You have two lines that presumably lie in each line in the planes you want. Any point on the line will do.
 

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