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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find parametric equations for the line through the point (0, 1, 2) that is parallel to the plane x + y + z = 2 and perpendicular to the line x = 1 + t, y = 1 − t, z = 2t.

To determine

To find: The parametric equations for the line through the point (0,1,2) , parallel to the plane x+y+z=2 , and perpendicular to the line x=1+t,y=1t,z=2t .

Explanation

Formula:

Write the expression to find the parametric equations for a line through the point (x0,y0,z0) and parallel to the direction vector a,b,c .

x=x0+at,y=y0+bt,z=z0+ct (1)

The direction vector of required line (a) is the cross product of normal vector of parallel plane and direction vector of perpendicular line.

Consider the normal vector for the parallel plane is n and direction vector for the perpendicular line is v .

Write the expression to find direction vector to the required line.

a=n×v

The expression is also written as follows.

a=|ijka1b1c1a2b2c2| (2)

Write the equation of plane which is parallel to the line.

x+y+z=2

Write the normal vector from the plane.

n=1,1,1

The line which is perpendicular to the required is in the form of parametric equations.

Write the parametric equations of the perpendicular line as follows.

x=1+t,y=1t,z=2t (3)

Compare equation (3) with (1) and write the direction vector v of perpendicular line as follows.

v=1,1,2

Calculation of direction vector of required line a :

Substitute 1 for a1 , 1 for b1 , 1 for c1 , 1 for a2 , 1 for b2 , and 2 for c2 in equation (2),

a=|ijk111

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Chapter 12 Solutions

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