Need help finding equation of a plane

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SUMMARY

The equation of a plane containing the point (0,1,3) and the line defined by the vector equation (x,y,z) = (-1,0,-2) + t(1,-3,-1) can be derived using the direction vector (1,-3,-1). To uniquely specify the plane, one must utilize the point-normal form of the plane equation, which requires a normal vector in addition to a point on the plane. The discussion emphasizes the importance of understanding the vector equation of a plane rather than jumping directly to the standard form Ax+By+Cz+D=0.

PREREQUISITES
  • Understanding of vector equations in three-dimensional space
  • Knowledge of the point-normal form of a plane equation
  • Familiarity with direction vectors and their role in defining planes
  • Basic algebraic manipulation skills for rearranging equations
NEXT STEPS
  • Study the derivation of the point-normal form of a plane equation
  • Learn how to calculate a normal vector from given points and direction vectors
  • Explore examples of finding equations of planes given points and lines
  • Practice converting between vector equations and standard forms of plane equations
USEFUL FOR

Students studying geometry, particularly those tackling problems involving planes in three-dimensional space, as well as educators looking for clear examples of plane equations in vector calculus.

sk_88
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Homework Statement



Hi,

So my question is:

Find the equation of the plane which contains the point (0,1,3) and the line:

(x,y,z) = (-1,0,-2) + t(1,-3,-1)

It has to be in the form Ax+By+Cx=D. D has to be positive

Homework Equations



see above

The Attempt at a Solution



the only thing I know right now is that the plane has the direction vector of (1,-3,-1) because this plane is parallel to the line. And that could possibly be vital to finding the solution. But I have absolutely no idea where to go from there. So I hope you guys can help me out with this, thanks!
 
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You know one direction vector, that is right.
This alone however is not enough to specify the plane you're lookng for.

What do you need to uniquely specify a plane?

For a line for example, two points would do.
 
See Pere Callahan's Post.
Addition:
1) Don't go straight for Ax+By+Cz+D = 0
2) hint: you need to know what's the vector equation of a plane
 

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