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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find the point at which the line intersects the given plane.

47. 5x = y/2 = z + 2; 10x − 7y + 3z    + 24 =     0

To determine

To find: The point at which the line with symmetric equations 5x=y2=z+2 intersects the plane 10x7y+3z+24=0 .

Explanation

The point of intersection of the line and the plane is determined by substituting the value of the scalar parameter (t) in the parametric equations of the line.

The scalar parameter is determined by substituting the parametric equations in the the equation of the plane.

Write the expressions for the symmetric equations for a line through the point (x0,y0,z0), parallel to the direction vector a,b,c .

xx0a=yy0b=zz0c (1)

Write the symmetric equations of the line as follows:

5x=y2=z+2

Rewrite the symmetric equations as follows:

x0(15)=y02=z(2)1 (2)

Compare equation (2) with (1) and write the point in the line and direction vector (a) of a line as follows.

(x0,y0,z0)=(0,0,2)a=15,2,1

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

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