# Equation of a plane involving a known line and a known point

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In summary, to find the equation of a plane that contains a given line and point, you must first determine the vector a that is parallel to the line and the point B(1,0,1). Then, find a vector n that is orthogonal to both a and the vector formed by the point B and the origin. Finally, use the point B and the vector n to form the equation of the plane. It is also important to check the validity of the normal vector you find through the cross product of a and b.
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## Homework Statement

Find the equation of the plane which contains the line r(t) = (2-t, 1+t, t) and the point (1,0,1).

## The Attempt at a Solution

Since r(t) = (2-t, 1+t, t), we know that vector a, which is parallel to the line, is (-1, 1, 1).
I'm assuming that since the point is (1,0,1), that means if we look to that point from the origin (0,0,0) we will have vector b = (1,0,1).
Okay so with two vectors a and b, I can find a vector n such that n is orthogonal to vectors a and b. So a x b = (1-0,1+1,0-1) = (1,2-1) = vector n.
The scalar equation of a plane is a(x-x0) + b(y-y0 + c(z-z0). So x + 2y -z = 0...

But somehow I have a feeling that my result is wrong. Is there supposed to be a "y" at all in the equation for this plane? Please advise. Thanks in advance!

The point B(1, 0, 1) is in the plane, but the vector OB isn't. From your equation for the line you can easily find two points in the plane, say (2, 1, 0) and (1, 2, 1), just by supplying values of t.

Form vectors between one of the points and each of the other two. Now cross these vectors to get a vector normal to both of these. Use this normal vector and anyone of the points to get the equation of the plane.

You didn't show your work in getting the cross product a X b, so the normal you show is somewhat suspect.

## What is an equation of a plane involving a known line and a known point?

An equation of a plane involving a known line and a known point is a mathematical representation that describes the relationship between a line and a point in a three-dimensional space.

## How do you find the equation of a plane using a known line and a known point?

To find the equation of a plane involving a known line and a known point, you can use the point-normal form of the equation, which is (x - x0, y - y0, z - z0) · n = 0, where (x0, y0, z0) is the known point and n is the normal vector of the plane.

## What is the normal vector of a plane?

The normal vector of a plane is a vector that is perpendicular to the plane and determines its orientation. It is represented by a set of three numbers (a, b, c) and is used in the point-normal form of the equation of a plane.

## Can the equation of a plane involving a known line and a known point have multiple solutions?

No, the equation of a plane involving a known line and a known point will only have one solution. This is because a plane is uniquely defined by a line and a point, and the equation represents the relationship between these two elements.

## What is the significance of the equation of a plane involving a known line and a known point in mathematics?

The equation of a plane involving a known line and a known point is significant in mathematics as it allows for the representation and analysis of relationships between lines and points in a three-dimensional space. It is also used in various applications, such as computer graphics, engineering, and physics, to solve problems involving planes.

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