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

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In summary, to find the equation of the plane through the given line and parallel to the other line, you need to find a normal vector that is perpendicular to both lines. This can be done by using the properties of vector products. Additionally, any point on the first line can be used to obtain the vector equation of the plane. From there, you can convert it to a cartesian equation if needed.
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nk735
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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.
 
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  • #2
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.
 

1. What is the equation of a plane through a given line?

The equation of a plane through a given line can be written in the form of Ax + By + Cz = D, where A, B, and C are the coefficients of the variables x, y, and z, and D is a constant. This equation represents all the points that lie on the plane, and the line must satisfy this equation as well.

2. How do you solve for the equation of a plane through a line with given points?

To solve for the equation of a plane through a line with given points, you first need to find the normal vector of the plane. This can be done by finding the cross product of two vectors that lie on the plane. Once you have the normal vector, you can substitute the coordinates of the given points into the equation Ax + By + Cz = D to find the values of A, B, C, and D.

3. What is the significance of the normal vector in the equation of a plane?

The normal vector is perpendicular to the plane, and it determines the orientation of the plane. It is also used to find the distance from a point to the plane, and it is crucial in solving problems involving planes, such as finding the intersection of two planes.

4. Can the equation of a plane through a line be written in different forms?

Yes, the equation of a plane through a line can also be written in the form of (x-x0, y-y0, z-z0) · n = 0, where (x0, y0, z0) represents a point on the plane and n is the normal vector. This form is known as the vector equation of a plane.

5. How is the equation of a plane through a line related to the vector equation of a plane?

The equation Ax + By + Cz = D and the vector equation (x-x0, y-y0, z-z0) · n = 0 are equivalent and can be converted to each other. The coefficients A, B, and C in the first equation represent the components of the normal vector in the second equation, and D represents the distance from the origin to the plane.

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