The following pairs of lines are not parallel Find the dista

The following pairs of lines are not parallel. Find the distance between each pair of lines. l_1: r(t) = (2,5,0) +t(0,0,1) and l_2: r(t) = (-1,-1,8) +t(-1,1,0) l_1: r(t) = (-1,3,4) +t(1,1, 2) and l_2: r(t) = (1,0,-1) +t(3,5,8) l_1: r(t) = (7,-4,-3) +t(1, 2, 3) and l_2: r(t) = (42,0,0) +t(-5,1, 2)

Solution

A) given l₁ = r(t)= (2,5,0) + t( 0,0,1)

l₂ =r(t)= (-1,-1,8) + t(-1,1,0)

the cross product of (0,0,1) and ( -1,1,0) = (-1, -1 ,0)

NOTE : the cross product of (a1, a2, a3 ) and ( b1, b2, b3)

=( (a2.b3- a3 .b2), (a3.b1 - a1.b3), (a1.b2 - a2.b1))

distance of L1, L2 is d(L1,L2) = |(3,6,-8) . (-1,-1,0)| / |(-1, 1, 0) |

   = 9 / square root(2)

B) L1 : r(t) =(-1,3,4) + t( 1,1,2)

L2 : r(t) = ( 1,0,-1) + t( 3,5,8)

the cross product of ( 1,1,2) and (3,5,8) is = ( -5,-2 ,2)

the diatance between L1 and L2 is = |(-2, 3, 5) . ( -5, -2 , 2)| / |(-5,-2, 2)|

   = |10-6+10| / | sqrt(25+4+4)|

   = 14 / sqrt(33)

C) given L1 : r(t) =( 7,-4 ,-3) + t(1,2,3)

L2 : r(t) = (42,0,0) + t( -5,1,2)

the cross product of (1,2,3) and ( -5, 1, 2) is = (1, -18, 11)

the distance between L1 and L2 is d(L1,L2) = |(-35, -4, -3 ) . (1, -18, 11)| / |( 1, -18, 11)|

   = 4 / sqrt(429)

D) all the above given pair of lines are skew.