   Chapter 11.5, Problem 75E

Chapter
Section
Textbook Problem

# Intersection of PlanesIn Exercises 65–68, (a) find the angle between the two planes and (b) find a set of parametric equations for the line of intersection of the planes. 3 x + 2 y − z = 7 x − 4 y + 2 z = 0

(a)

To determine

To calculate: planes 3x+2yz=7 and x4y+2z=0 are given. Find the angle between the planes.

Explanation

Given:

The equation of the two planes are:

3x+2yz=7

And,

x4y+2z=0

Formula used:

The angle between the two planes is given by:

cosθ=|n1n2|n1n2

Calculation:

As we know any two distinct planes in three-space are either parallel or intersect in a line.

Now The angle between two intersecting planes can be determined from the angle between their normal vectors.

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 planes are:

3x+2yz=7

And,

x4y+2z=0

Coordinates of the normal vector are the coefficients of x,y,z in the standard equation of plane

(b)

To determine

To calculate: Find the set of parametric equations for the line of intersection of planes 3x+2yz=7 and x4y+2z=0.

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