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 11: University Calculus: Early Transcendentals
Uploaded: 2020-04-22T16:59:49-08:00
Duration: 2 min 12 s
Channel: Linh Vu
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 In this question we need to find the equation of the plan such that it going through the point P

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

Key Concepts

Plane Equation

A plane in three-dimensional space can be represented by an equation of the form Ax + By + Cz + D = 0, where (A, B, C) is a normal vector perpendicular to the plane. This representation allows one to determine whether a point lies on the plane and to analyze the spatial relationships between different planes, such as parallelism and perpendicularity.

Normal Vector

The normal vector is a vector that is perpendicular to a plane. In the context of plane equations, the coefficients of x, y, and z correspond to the components of the normal vector. Understanding the normal vector is crucial because it governs the orientation of the plane in space and is used to determine relationships like orthogonality or angle between planes.

Dot Product

The dot product is an algebraic operation that takes two vectors and returns a scalar. It is particularly useful in determining the angle between two vectors: if the dot product is zero, the vectors are orthogonal. In problems involving perpendicular planes, the dot product helps verify that certain vectors, such as a plane's normal vector and a given direction vector, are perpendicular.

Cross Product

The cross product of two vectors in three-dimensional space results in a third vector that is perpendicular to both of the original vectors. This operation is essential when finding a normal vector to a plane defined by two non-parallel direction vectors, such as the vector connecting two points in the plane and another vector determined from perpendicularity conditions.

Point in Plane

A point in three-dimensional space is said to lie on a plane if its coordinates satisfy the plane's equation. Identifying points on a plane is fundamental to defining the plane itself, as well as for solving for unknown parameters in the plane's equation, especially when the plane is determined by given points and additional geometric conditions.

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