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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find an equation of the plane.

26. The plane through the point (2, 0, 1) and perpendicular to the line x = 3t, y = 2 − t, z = 3 + 4t

To determine

To find: An equation of the plane that passes through the point (2,0,1) and perpendicular to the line x=3t,y=2t,z=3+4t .

Explanation

The parametric equations of the line are written as follows.

x=3t,y=2t,z=3+4t

Rewrite the parametric equations of the line as follows.

x=0+3t,y=2t,z=3+4t (1)

Formula:

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

a(xx0)+b(yy0)+c(zz0)=0 (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 (1) and writ the direction vector of the line (v) .

v=3,1,4

As the plane is perpendicular to the line, the direction of line 3,1,4 is considered as normal vector to the plane

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