Any help with this question would be greatly appreciated Tha

Any help with this question would be greatly appreciated! Thanks!

Let L_1 be the line passing through the points Q_1=(-1, 1, -1) and Q_2=(5, -5, -3) and let L_2 be the line passing through the point P_1=(-19, 13, -2) with direction vector d^rightarrow=[l, 1, -1]^T. Determine whether L_1 and L_2 intersect. If so, find the point of intersection Q. If not, find a value for the z-coordinate of P_1 so the resulting lines do intersect.

Solution

(-1 , 1, -1) ( 5, -5, -3)

Dierction vectors : ( 6 , -6, -2)

L1 : (-1, 1, -1) + s( 6 , -6 , -2)

(-19, 13, -2) and (1,1,-1)^T

L2 : (-19, 13, -2) + t(1, 1, -1)

Equating the both : -1 +6s = -19 +t ;

                                1 -6s = 13 +t

we get -2 +12s = -32 ; 12s = -30 ; s= -30/12

s = - 15/6 ; 1 -6(-15/6) = 13 +t

16 = 13 +t ; t = 3

So, Point of intersection : (-19, 13, -2) + t(1, 1, -1) = ( -16 , 16 , -5)