Answer to Question #135215 in Analytic Geometry for Yolanda

Question

whats the relationship between (2,1,2) and the line L represented by parametric equations x=1+2t,y=2+t,z=3-2t where t is an element of R

Answer 1

The parametric equations x=1+2t, y=2+t, z=3-2t

Write the vector equation of the line:

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

A plane that is perpendicular to this line will have the general equation:

2x + y – 2z = c

Substitute the point (2, 1, 2) and then solve for c:

2(2) + 1 – 2(2) = c

C = 1

The plain 2x + y – 2z = 1 contains the point (2, 1, 2) and is perpendicular to the line:

To find the point where the line intersects the plane, substitute the parametric equations of the line into the equation of the plane:

2(1 + 2t) + 1(2+t) + 2(3-2t) = 1

2 + 4t + 2 + t +6 – 4t = 1

10 + t = 1

t = -9

x= 1 + 2(-9)

x= -17

y = 2 + - 9

y= -7

z = 3 – 2(-9)

z= 21

The line intersects the plane at the point (-17, -7, 21)

Check 2x + y – 2z = 1

2(-17) – 7+2(21) = 1

The vector, v, from the given point to the intersection point is:

V= (-17 – 2)i + (-7 - 1)j + (21  - 2)k

V= -19i -8j+19k

The vector equation of the line is:

(x, y, z) = (2,1,2) + r (-19i -8j+19k)

This is the relationship between x=1+2t, y=2+t, z=3-2t and the point (2,1,2):

(x, y, z) = (2,1,2) + r (-19i -8j+19k)