Is the line parallel to the plane?

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SUMMARY

The line defined by the parametric equations x=1-2t, y=2+5t, z=-3t is not parallel to the plane described by the equation 2x+y-z=8. The determination was made using the dot product of the line's directional vector and the plane's normal vector. If the line were parallel to the plane, it would be perpendicular to the normal vector of the plane. Additionally, if the line is not parallel, it will intersect the plane at some point.

PREREQUISITES
  • Understanding of parametric equations
  • Knowledge of vector operations, specifically dot products
  • Familiarity with the concept of normal vectors in geometry
  • Basic principles of plane equations in three-dimensional space
NEXT STEPS
  • Study the properties of normal vectors in relation to planes
  • Learn how to calculate intersections of lines and planes
  • Explore vector calculus and its applications in geometry
  • Review the concept of parallelism in three-dimensional space
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Students and professionals in mathematics, physics, and engineering who are working with geometric concepts involving lines and planes, particularly in three-dimensional space.

cyt91
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Is the line x=1-2t,y=2+5t,z=-3t parallel to the plane 2x+y-z=8?

I've used the approach of finding the dot product of the directional vector of the line and the vector normal to the plane in this question. The line is not parallel to the plane.
But I'm not too sure if my working is correct. Can anyone please check my working?

It's here:

https://skydrive.live.com/?cid=6b04...751C72E14AD&id=6B041751C72E14AD!160&sc=photos

Thank you.
 
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It is correct. But you need to write a sentence for explanation, that if it was parallel with the plane than it would be perpendicular to the normal vector of the plane.

ehild
 
Ok. Thanks.
 
Also, if the line were not parallel to the plane, it would intersect the plane at some point in the plane. Does it?
 

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