3D Geometry: Intersection of Lines & Parallel Lines

Question

Question:

Find the equations o the line that intersects the lines

2x + y - 1 = 0 = x - 2y + 3z (say 1)

3x - y + z - 2 + 2 = 0 = 4x + 5y - 2x - 3 (say 2)

and is parallel to the line x/1 = y/2 = z/3 (say 3)

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I noted that the RHS of (2) did not have a z coefficient, which I'm not sure is a typo, since this is how it was given in the question.

I had 2 approaches for this questions as follows:

(1) Equations of general lines passing through above are given by,

2x + y - 1 +k(x - 2y + 3z)----- (4) and

3x - y + z - 2 + 2 + r(4x + 5y - 2x - 3) ----(5) for some real number k &r

Let A & B be the points that the required line intersects (4) & (5).

Also, the required line is given to be parallel to the line x/1 = y/2 = z/3

So due to parallelism, the direction ratios of (3) & (4) and (3) & (5) should be in proportional at the points of intersection A & B.

i.e. (2+k)/1 = (1-2k)/2 = 3k/3 ---(6) and

(3+2r)/1 = (5r-1)/2 = 1/3 ---(7), which gives,

2+k=k from (6), which gives 2=0(??????)

Am I taking the right approach here?

Answer 1

I do think you can approach better, once the givens are typed coherently.

two planes intersect to form L1, put into form x1=a₁t+x₁, y1=b₁t+y₁, z1=c₁t+z₁

for the other two planes and L2, do the same with subscipts now =2 and use variable s instead of t

find a sure fire way to find L1 and L2. using simultaneous equations and putting z=0, then z=1

for the two planes in question will give points and slope ratios, but some way need to find L1 and L2.

Now the idea is to say x2-x1=f,,,y2-y1=2f,,,,,z2-z1=3f

what is next is a matrix wijth the variable vector as (s,t,f)^T, the costant vector as (x₂-x₁,y₂-y₁,z₂-z₁)^T

and matrix

row1=a₂,,,a₁,,,-1

row2=b₂,,,b₁,,,-2

row3=c₂,,,c₁,,,,-3

Should find valid s,t,f splving matrix equation

use t to find point on L1, use direction vector given to make line that is solution.

With the garbling of the question (who has +2-2, or 4x-2x as part of equation)

it is not possible to confidently find solution specifically.