Equation of Plane Through Line: Solve w/ Points & Normal Vector

[nk735]

Homework Statement

Find the equation of the plane through the line; r=(a,b,c)+t(d,e,f), and parallel to the line; r=s(g,h,i)

(note: a-i are all real numbers - but I'm not telling what because I don't like people solving my problems, s and t are parametric variables)

Homework Equations

Ax+By+Cz=D

The Attempt at a Solution

- I realize I need a point on the plane and a normal vector to the plane

- if it's parallel to the line, then the normal to the line is the same as the normal to the plane. So I need to find the normal to the line... somehow?

- I'm not quite sure how to work with the information 'plane through the line', but I'm assuming this is supposed to somehow give me my point?

If someone could give me a push as to how to find those two pieces of information, I should have no problem finding the equation of the plane, thank-you.

 

[rohanprabhu]

As you said, you need a normal vector through the plane. One of the line passes through the plane and the other is parallel to it. As such, both the lines will be perpendicular to the plane. Hence, the normal to the plane would be a vector such that it is perpendicular to both the given lines. How can you get such a line? [Think in terms of products].

Secondly, every point on the first line lies on the plane. As such, any value for 't' would give you a point through the plane. Use this data alongwith the previously obtained data to get the vector equation of the plane. Then convert it to cartesian equation if required.