(b) Consider the three points A(2,3, 1), B(-3,2,2) and C(1, –1,3), as well as the plane P definedby r - 2y + 3z = -4.(i) Find the vector equation of the line through A and B.(ii) Show that this line does not intersect the plane P.(iii) Find the vector equation of the line through A and C. Find its intersection with P.

Answered: (b) Consider the three points A(2,3,…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.1: Length And Dot Product In R^n

Problem 17E: Consider the vector v=(1,3,0,4). Find u such that a u has the same direction as v and one-half of...

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(b) Consider the three points A(2,3, 1), B(-3,2,2) and C(1, –1,3), as well as the plane P defined
by r - 2y + 3z = -4.
(i) Find the vector equation of the line through A and B.
(ii) Show that this line does not intersect the plane P.
(iii) Find the vector equation of the line through A and C. Find its intersection with P.

Expert Solution

Step 1

Hello. Since your question has multiple sub-parts, we will solve first three sub-parts for you. If you want remaining sub-parts to be solved, then please resubmit the whole question and specify those sub-parts you want us to solve.

The straight line joining points is A (2,3,1) and B(-3,2,2) is

$\frac{x - 2}{2 - (- 3)} = \frac{y - 3}{3 - 2} = \frac{z - 1}{1 - 2} = r$

$\frac{x - 2}{5} = \frac{y - 3}{1} = \frac{z - 1}{- 1} = r$

Step 2

Plane P: x-2y+3z=4

Checking the point of intersection:

x=5r+2,y=r+3,z=1-r

5r+2-2(r+3)+3(1-r)=4

2-6+3=4

-1 is not equal to 4.

Therefore, Plane does not intersect line.

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