1. ## 2(cd) intersecting lines

wait wait!
I got values of μ and λ from the earlier post.
I compared i and j components and substituted back into equation 1

r1=[j+ak] +λ[2i+4j-3k]
r2=[4i+3j-6k]+μ[i+5j+k]

I need to rewrite these in parametric form

λi+(1+4λ)j+(3-3λ)k=(4+μ)i+(3+5μ)j+(6+μk)

comparing i components:
(1)
λ=4+μ

comparing j components:
(2)
1+4λ=3+5μ
λ=(2+5μ)/4

(1)=(2)
4+μ=(2+5μ)/4
16+4μ=2+5μ
μ=14

sub into (1)
λ=4+μ
λ=4+14
λ=18

Different values so the lines are skew

d) Find a value of a for which the lines intersect and state the coordinates of the point of intersection.
r1=2λi+(4λ+1)j+(a-3λ)k

r2=(μ+4)i+(5μ+3)j+(μ-6)k

I need to find the value of a for which the lines intersect and state the coordinates of point of intersection.

λ=18
μ=14

LHS a-3λ=μ-6
a-3(18)=14-6
a-54=8
a=62

r2=4i+3j-6k+μ[i+5j+k]
r2=4i+3j-6k+14[i+5j+k]
r2=4i+3j-6k+14i+70j+14k
18i+73j-8k

coordinates of point of intersection are:
(18, 73, -8)

How's that?

2. wait wait!
I got values of μ and λ from the earlier post.
I compared i and j components and substituted back into equation 1

r1=[j+ak] +λ[2i+4j-3k]
r2=[4i+3j-6k]+μ[i+5j+k]

I need to rewrite these in parametric form

2λi+(1+4λ)j+(3-3λ)k=(4+μ)i+(3+5μ)j+(-6+μ)k

comparing i components:
(1)
2λ=4+μ
λ=(4+μ)/2

comparing j components:
(2)
1+4λ=3+5μ
λ=(2+5μ)/4

(1)=(2)
(4+μ)/2=(2+5μ)/4
16+4μ=4+10μ
μ=2

sub into (1)
λ=(4+μ)/2
λ=(4+2)/2
λ=3

Different values so the lines are skew

d) Find a value of a for which the lines intersect and state the coordinates of the point of intersection.
r1=2λi+(4λ+1)j+(a-3λ)k

r2=(μ+4)i+(5μ+3)j+(μ-6)k

I need to find the value of a for which the lines intersect and state the coordinates of point of intersection.

λ=3
μ=2

LHS a-3λ=μ-6
a-3(3)=2-6
a-9=-4
a=5

r2=4i+3j-6k+μ[i+5j+k]
r2=4i+3j-6k+2[i+5j+k]
r2=4i+3j-6k+2i+10j+2k
6i+13j-4k

coordinates of point of intersection are:
(6, 13, -4)

Somebody pleeeease tell me this is correct?