Precalculus with Limits: A Graphing Approach
Precalculus with Limits: A Graphing Approach
7th Edition
ISBN: 9781305071711
Author: Ron Larson
Publisher: Brooks/Cole
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Chapter 10.4, Problem 50E

(a)

To determine

To find: The angle between the two planes x−3y+2z=0 and 3x+2y−5z=0 .

(a)

Expert Solution
Check Mark

Answer to Problem 50E

The angle between the two planes x−3y+2z=0 and 3x+2y−5z=0 is 124.30° .

Explanation of Solution

Given information:

The equation of two planes x−3y+2z=0 and 3x+2y−5z=0 .

Calculation:

Compare the equation of the plane with ax+by+cz=d .

So, the normal vector for plane x−3y+2z=0 is 〈1,−3,2〉 and normal vector for plane 3x+2y−5z=0 is 〈3,2,−5〉 .

The angle between the two planes is equal to the angles between their normal vectors.

Use the formula for angle between two vectors u and v .

  cosθ=u⋅v‖u‖‖v‖

The dot product of both the vectors 〈1,−3,2〉 and 〈3,2,−5〉 .

  〈1,−3,2〉⋅〈3,2,−5〉=1×3+(−3)×2+2×−5=3−6−10=−13

Use the formula for magnitude of vector 〈1,−3,2〉 .

  ‖〈1,−3,2〉‖=(1)2+(−3)2+(2)2=1+9+4=14

Use the formula for magnitude of vector 〈3,2,−5〉 .

  ‖〈3,2,−5〉‖=(3)2+(2)2+(−5)2=9+4+25=38

Substitute all the values in the formula for angle.

  cosθ=〈1,−3,2〉⋅〈3,2,−5〉‖〈1,−3,2〉‖‖〈3,2,−5〉‖=−1314×38=−13133266

The solution for the trigonometric equation cosθ=−13133266 is θ=124.30° .

Therefore, the angle between the two planes x−3y+2z=0 and 3x+2y−5z=0 is 124.30° .

(b)

To determine

To find: The parametric equations of the line of intersection for the planes x−3y+2z=0 and 3x+2y−5z=0 .

(b)

Expert Solution
Check Mark

Answer to Problem 50E

The parametric equations of the line of intersection of the planes x−3y+2z=0 and 3x+2y−5z=0 is x=t , y=t and z=t .

Explanation of Solution

Given information:

The equation of two planes x−3y+2z=0 and 3x+2y−5z=0 .

Calculation:

Multiply the first equation of plane by 3 and subtract from the second plane.

  3x+2y−5z−3(x−3y+2z)=03x+2y−5z−3x+9y−6z=011y−11z=0y=z

Substitute y=z in the equation of plane x−3y+2z=0 .

  x−3y+2z=0x−3z+2z=0x−z=0x=z

Finally assume that t=z .

Therefore, the parametric equations of the line of intersection of the planes x−3y+2z=0 and 3x+2y−5z=0 is x=t , y=t and z=t .

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Chapter 10.4, Problem 49E
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Chapter 10.4, Problem 51E

Chapter 10 Solutions

Precalculus with Limits: A Graphing Approach

Ch. 10.1 - Prob. 1ECh. 10.1 - Prob. 2ECh. 10.1 - Prob. 3ECh. 10.1 - Prob. 4ECh. 10.1 - Prob. 5ECh. 10.1 - Prob. 6ECh. 10.1 - Prob. 7ECh. 10.1 - Prob. 8ECh. 10.1 - Prob. 9ECh. 10.1 - Prob. 10E
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Prob. 12ECh. 10.4 - Prob. 13ECh. 10.4 - Prob. 14ECh. 10.4 - Prob. 15ECh. 10.4 - Prob. 16ECh. 10.4 - Prob. 17ECh. 10.4 - Prob. 18ECh. 10.4 - Prob. 19ECh. 10.4 - Prob. 20ECh. 10.4 - Prob. 21ECh. 10.4 - Prob. 22ECh. 10.4 - Prob. 23ECh. 10.4 - Prob. 24ECh. 10.4 - Prob. 25ECh. 10.4 - Prob. 26ECh. 10.4 - Prob. 27ECh. 10.4 - Prob. 28ECh. 10.4 - Prob. 29ECh. 10.4 - Prob. 30ECh. 10.4 - Prob. 31ECh. 10.4 - Prob. 32ECh. 10.4 - Prob. 33ECh. 10.4 - Prob. 34ECh. 10.4 - Prob. 35ECh. 10.4 - Prob. 36ECh. 10.4 - Prob. 37ECh. 10.4 - Prob. 38ECh. 10.4 - Prob. 39ECh. 10.4 - Prob. 40ECh. 10.4 - Prob. 41ECh. 10.4 - Prob. 42ECh. 10.4 - Prob. 43ECh. 10.4 - Prob. 44ECh. 10.4 - Prob. 45ECh. 10.4 - Prob. 46ECh. 10.4 - Prob. 47ECh. 10.4 - Prob. 48ECh. 10.4 - Prob. 49ECh. 10.4 - Prob. 50ECh. 10.4 - Prob. 51ECh. 10.4 - Prob. 52ECh. 10.4 - Prob. 53ECh. 10.4 - Prob. 54ECh. 10.4 - Prob. 55ECh. 10.4 - Prob. 56ECh. 10.4 - Prob. 57ECh. 10.4 - Prob. 58ECh. 10.4 - Prob. 59ECh. 10.4 - Prob. 60ECh. 10.4 - Prob. 61ECh. 10.4 - 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