   Chapter 12.5, Problem 30E

Chapter
Section
Textbook Problem

# Find an equation of the plane.30. The plane that contains the line x = 1 + t, y = 2 − t, z = 4 − 3t and is parallel to the plane 5x + 2y + z = 1

To determine

To find: An equation of the plane that contains the line x=1+t,y=2t,z=43t and parallel to the plane 5x+2y+z=1 .

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)

The normal vector of the plane 5x+2y+z=1 is 5,2,1 .

As the two planes are in parallel, the normal vectors of both the planes are equal.

Therefore, the normal vector of the plane that contains the line x=1+t,y=2t,z=43t is also 5,2,1 .

n=5,2,1

The parametric equations of the line are written as follows.

x=1+t,y=2t,z=43t (2)

Write the expressions for 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 (3)

Compare equation (3) with equation (2) and write the point at which the line passes

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