   Chapter 12.5, Problem 48E

Chapter
Section
Textbook Problem

# Where does the line through (−3, 1, 0) and (−1, 5, 6) intersect the plane 2x + y − z = −2?

To determine

To find: The point at which the line through the points (3,1,0) and (1,5,6) intersects the plane 2x+yz=2 .

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 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)

Write the expression to find the direction vector from the point P(x1,y1,z1) to Q(x2,y2,z2) .

PQ=(x2x1),(y2y1),(z2z1) (2)

Consider the direction vector from the point (3,1,0) to (1,5,6) to be (a) .

Calculation of direction vector a :

Substitute 3 for x1 , 1 for y1 , 0 for z1 , 1 for x2 , 5 for y2 , and 6 for z2 in equation (2),

a=[1(3)],(51),(60)=2,4,6

Consider the line passes through the point (3,1,0) as (x0,y0,z0) and the direction vector 2,4,6 is a,b,c

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Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach 