Find an equation for the plane containing the point (2,3,4) and the line x=1+2 t, y=3-t, z=4+t.

Name: Problem 56, Chapter 17: Calculus Multivariable
Uploaded: 2023-06-21T04:42:08-08:00
Duration: 5 min 37 s
Channel: Khushbu Rani
Description: Find an equation for the plane containing the point (2,3,4) and the line x=1+2 t, y=3-t, z=4+t.

Find an equation for the plane containing the point (2,3,4)...

Key Concepts

Point-Normal Form of a Plane

This is a way to express the equation of a plane using a point in the plane and a normal vector to the plane. The form is based on the idea that the dot product of the normal vector with any vector lying on the plane (originating from the known point) is zero. This concept is fundamental in converting geometric descriptions into algebraic equations for planes.

Parametric Representation of a Line

This concept involves expressing a line in three-dimensional space using parameters. Instead of using a standard y = mx + b form (which is limited to two dimensions), the line is described by equations for x, y, and z in terms of a parameter t. This allows for a clear depiction of the line's direction and position in space, which is essential for understanding how the line interacts with other geometric entities like planes.

Direction Vector

The direction vector of a line is a vector that indicates the line's orientation in space. It is derived directly from the coefficients of the parameter in the line's parametric equations. In problems involving planes and lines, the direction vector is key because it represents a direction that lies within the plane when the line is contained in the plane.

Vector Subtraction

Vector subtraction is a fundamental operation in vector algebra used to obtain the vector between two points. In the context of finding a plane, subtracting the coordinates of two points that lie on the plane yields a vector that lies within the plane. This is crucial for constructing additional vectors in the plane that can be used to determine its orientation.

Cross Product

The cross product is an operation on two vectors in three-dimensional space that results in a third vector perpendicular to the plane containing the original vectors. By taking the cross product of two non-parallel vectors lying in a plane, one can obtain a normal vector, which is essential for defining the plane's equation.

Normal Vector

A normal vector is a vector that is perpendicular to a given plane. It plays a central role in the point-normal form of a plane equation. Once the normal vector is determined (often via the cross product of two vectors in the plane), it can be used along with a point on the plane to uniquely define the plane's equation in space.

Transcript

00:01 As the equation say that find an equation for the plane containing the point 2, 3, 4.

00:07 Let us say this point is l and the line x equal to 1 plus 2t, y equal to 3 minus t and z equal to 4 plus t.

00:21 So let us first find out the cartesian equation of the line.

00:25 This is parametric equation.

00:26 So from here t will be x minus 1 by 2 and t will be 3 minus y and from here t will be z minus 4.

00:42 If we equate all the values of t it will be x minus 1 by 2, 3 minus y, z minus 4.

00:52 This could be written as x minus 1 by 2 and i will be dividing with minus 1, y minus 3 and z minus 4 by 1.

01:02 I have to compare this with x minus x1 by a, a1 let us say and y minus y1 by b and z minus z1 by c.

01:19 Let us say i will write this only, a, b and c.

01:24 So the point on the line or the line passes through minus 1, minus 3, minus 4.

01:32 Let us write point a and this is point p, not l.

01:38 And this point a and this point p, that is these two points will lie on the plane.

01:44 Now i will be finding out the equation of the normal to the plane because we have equation of, required equation of plane that will be a into x minus x1 plus b into y minus y1 plus c into z minus z1 equal to 0 where a, b, c are the direction cosines of the normal perpendicular to the plane or normal to the plane.

02:21 Both are one and the same thing...

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