Find a plane through the points P1(1,2,3), P2(3,2,1) and perpendicular to the plane 4 x-y+2 z=7.

Name: Problem 32, Chapter 12: Thomas Calculus
Uploaded: 2019-08-02T22:38:09-08:00
Duration: 3 min 15 s
Channel: Tyler Gaona
Description: Find a plane through the points P1(1,2,3), P2(3,2,1) and perpendicular to the plane 4 x-y+2 z=7.

step 1 So here we want to find a plane that passes through these two given points and is perpendicular

Find a plane through the points P1(1,2,3), P2(3,2,1) and...

Key Concepts

Plane Equation

The plane equation is usually expressed in the form n · (r - r0) = 0, where n is the normal vector to the plane and r0 is a point on the plane. This formulation allows one to uniquely define a plane in three-dimensional space once a point and a normal are known.

Normal Vector

A normal vector is a vector that is perpendicular to every direction lying in the plane. It is a key element in determining the orientation of the plane and is used directly in the plane’s equation.

Direction Vectors

Direction vectors are vectors that lie within the plane. They can be obtained by taking the difference between any two distinct points on the plane. These vectors are essential in defining the plane's geometry and are used to compute the normal vector via the cross product.

Cross Product

The cross product of two non-parallel vectors results in a vector that is perpendicular to both. In the context of planes, it is used to find a normal vector to the plane when two independent direction vectors lying in the plane are available.

Perpendicular Planes Condition

When a plane is perpendicular to another plane, there is a specific relationship between their normals. Often, the normal vector of one plane lies within the perpendicular plane, or equivalently, certain vectors lying in the plane must be perpendicular to the normal of the given plane. This condition is used to establish the orientation of the plane relative to another.

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