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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find an equation of the plane.

31. The plane through the points (0, 1, 1), (1, 0, 1), and (1, 1, 0)

To determine

To find: An equation of the plane that passes through the points (0,1,1) , (1,0,1) and (1,1,0) .

Explanation

Formula:

Write the expression to find equation of the plane through the point P0(x0,y0,z0) with normal vector n=a,b,c as follows.

a(xx0)+b(yy0)+c(zz0)=0 (1)

Write the expression to find vector from the point P(x1,y1,z1) to Q(x2,y2,z2) .

PQ=(x2x1),(y2y1),(z2z1) (2)

Consider the vector from the point (0,1,1) to (1,0,1) is (a) and the vector from the point (0,1,1) to (1,1,0) is (b) .

Calculation of vector a :

Substitute 0 for x1 , 1 for y1 , 1 for z1 , 1 for x2 , 0 for y2 , and 1 for z2 in equation (2),

a=(10),(01),(11)=1,1,0

Calculation of vector b :

Substitute 0 for x1 , 1 for y1 , 1 for z1 , 1 for x2 , 1 for y2 , and 0 for z2 in equation (2),

b=(10),(11),(01)=1,0,1

As both the vectors a and b lie on the plane, the cross product of a and b is the orthogonal of the plane and it is considered as normal vector.

Find the normal vector (n)

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Chapter 12 Solutions

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P-38RECh-12 P-1PCh-12 P-2PCh-12 P-5P

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