   Chapter 11.5, Problem 107E

Chapter
Section
Textbook Problem

# Finding a Point of Intersection Find the point of intersection of the line through (1, -3, 1) and (3, -4, 2) and the plane given by x – y + z = 2.

To determine

To calculate: The point of intersection of the plane xy+z=2 and the line passing through the points (1,3,1) and (3,4,2).

Explanation

Given:

The equation of the plane is xy+z=2 and a line is passing through the points (1,3,1) and (3,4,2).

Formula used:

Parametric equation of the line passing through the point (x1,y1,z1) is given by,

x=x1+at,y=y1+bt,z=z1+ct.

Calculation:

The line is passing through the points (1,3,1) and (3,4,2), so, the parametric equations of a line through two points P (1,3,1) and Q (3,4,2).

The direction vector for the line passing through P and Q is,

v=PQ=31,4+3,21=2,1,1

Thus, the direction number is,

a=2b=1c=1

Now, from the direction number and the with the point P (1,3,1) the parametric equation is;

x=1+2ty=3tz=1+t

Substitute x=

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