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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Comparing Planes In Exercises 13–22, determine whether the planes

a 1 x + b 1 y + c 1 z = d 1

and

a 2 x + b 2 y + c 2 z = d 2 are parallel, perpendicular, or neither. The planes are parallel when there exists a nonzero constant k such that a 1 = k a 2 , b 1 = k b 2 , and c 1 = k c 2 , and are perpendicular when a 1 a 2 + b 1 b 2 + c 1 c 2 = 0.

5 x 3 y + z = 4 , x + 4 y + 7 z = 1

To determine

Whether the equations of plane 5x3y+z=4 and x+4y+7z=1 are perpendicular or parallel or neither perpendicular nor parallel.

Explanation

Given Information:

The equations of plane are:

5x3y+z=4x+4y+7z=1

The standard equations of planes are a1x+b1y+c1z+d1=0 and a2x+b2y+c2z+d=20

Planes to be parallel if

a1a2=b1b2=c1c2=k …… (1)

Planes to be perpendicular

a1a2+b1b2+c1c2=0 …… (2)

Equation of given planes are

5x3y+z=4 …… (3)

x+4y+7z=1 …… (4)

Now, on comparing equation (3) with standard equation of plane a1x+b1y+c1z+d1=0:

a1=5, b1=3, c1=1 and d1=4

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