Parallel lines???

Question

Q1) Consider the lines x = (5,1,4) + td  and  x = (0,4,3) + se , where d = (0,-1,2)  and  e = (0,2,-4)

 

a) How do we know these two lines are parallel?

1) because d and e both have a zero first component

2) because d . e ≠ 0

3) because d . e = 0

4) because d = ke for some non-zero scalar k

 

b) Find the distance between these parallel lines

 

Answer 1

a)

d = (0,-1,2)  and  e = (0,2,-4)

d = (-1/2) e

When d=ke lines are parallel

Therefore,option 4 is correct.

b)

Calculate distance between the pralles lines

( x, y,z)= (5,1,4) + t (0,-1,2)

t = (x-5)/0 = (y-1)/-1 =(z-4)/2

( x, y,z) = (0,4,3) + s(0,2,-4)

s =  (x-0)/0 = (y-4)/2 =(z-3)/-2

A =(5,1,4)  B(0,4,3)

AB=(-5,3,-1)

u=(0,-1,2)

v =(0,2,-2)

Distance between the pralles lines,h =[AB u  v]/|u×v|

[AB u  v]=|-5  -3   -1 |

               | 0   1     2 |

               |0    2   -2  |

            = -5 (-2-4)+3(0-0)-1(0-0)

           =30

u×v= (0,-1,2)×(0,2,-2) 

     =(-2,0,0)

|u×v|=2

h =30/2

   =15