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High School Math Calculus PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)

The angle between the two planes x + y − z = 0 and 2 x − 5 y − z = 1 .

The angle between the two planes x + y − z = 0 and 2 x − 5 y − z = 1 .

PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)

PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)

8th Edition

ISBN: 9781337904285

Author: Larson

Publisher: CENGAGE L

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Question

Chapter 10.4, Problem 49E

(a)

To determine

To find: The angle between the two planes x+y−z=0 and 2x−5y−z=1 .

(a)

Expert Solution

Answer to Problem 49E

The angle between the two planes x+y−z=0 and 2x−5y−z=1 is 77.87° .

Explanation of Solution

Given information:

The plane are x+y−z=0 and 2x−5y−z=1 .

Calculation:

The angle between two planes is given by,

cosθ=|n1⋅n2|‖n1‖‖n2‖

From the planes,

n1=〈1,1,−1〉n2=〈2,−5,−1〉

Calculate the dot of n1 and n2 .

n1⋅n2=〈1,1,−1〉⋅〈2,−5,−1〉=2−5+1=−2|n1⋅n2|=2

Calculate ‖n1‖ and ‖n2‖ as follows.

‖n1‖=12+12+(−1)2=3‖n2‖=22+(−5)2+(−1)2=30

Substitute 2 for |n1⋅n2| , 3 for ‖n1‖ and 30 for ‖n2‖ in the equation cosθ=|n1⋅n2|‖n1‖‖n2‖ to find angle between the planes.

cosθ=23⋅30=0.21θ=cos−1(0.21)=77.87°

Therefore, the angle between the two planes x+y−z=0 and 2x−5y−z=1 is 77.87° .

(b)

To determine

To find: The parametric equation of their lines of intersection.

(b)

Expert Solution

Answer to Problem 49E

The parametric equations are x=17+6t , y=−17+t and z=7t .

Explanation of Solution

Given information:

The plane are x+y−z=0 and 2x−5y−z=1 .

Calculation:

To get the line that intersects, solve the two plane equations.

Substitute −y+z for x in the equation 2x−5y−z=1 .

2(−y+z)−5y−z=1−2y+2z−5y−z=17y=z−1y=−17+17z

Substitute t=z7 in the equation y=−17+17z .

y=−17+tz=7t

and,

x=17+6t

Therefore, the parametric equations are x=17+6t , y=−17+t and z=7t .

Chapter 10.4, Problem 48E

Chapter 10.4, Problem 50E

Chapter 10 Solutions

PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)

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