   Chapter 11.5, Problem 59E

Chapter
Section
Textbook Problem

# Comparing PlanesIn Exercises 69–74, determine whether the planes are parallel, orthogonal, or neither. If they are neither parallel nor orthogonal, find the angle between the planes. x − 3 y + 6 z = 4 5 x + y − z = 4

To determine
Two planes x3y+6z=4 and 5x+yz=4 are given. Find if the planes are orthogonal, parallel or neither. If neither orthogonal nor parallel then find the angle.

Explanation

Given:

The equation of the first planes is,

x3y+6z=4

And the equation of the second plane is,

5x+yz=4

Explanation:

For two planes x3y+6z=4 and 5x+yz=4

Angle between them is,

cosθ=|n1n2|n1n2

Now vectors n1 and n2 are normal to two intersecting planes,

Hence the angle θ between the normal vectors is equal to the angle between the two planes.

The equations of the first planes is:

x3y+6z=4

The equation of the second planes is,

5x+yz=4

The coordinates of the normal vector are the coefficients of x,y and z in the equation of the plane. Then,

The vector of the first plane is n1 then,

n1=1,3,6

The vector of the second plane is n2 then,

n2=5,1,1

Let us now, find the angle between the normal vectors

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