   Chapter 11.5, Problem 61E

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 − 5 y − z = 1 5 x − 25 y − 5 z = − 3

To determine
Whether the planes x5yz=1 and 5x25y5z=3 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,

x5yz=1

And the equation of the second plane is,

5x25y5z=3

Explanation:

For two planes x5yz=1 and 5x25y5z=3

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:

x5yz=1

And the equation of the second plane is,

5x25y5z=3

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

The normal vector of the first plane is n1 then,

n1=1,5,1

The normal vector of the second plane is n2 then,

n2=5,25,5

Let us now, find the angle between the normal vectors

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