Let Li be the line passing through the point P3 3 3 with dir

Let Li be the line passing through the point P.,=(3, -3, -3) with direction vector rightarrow d=[-2, -1,1], and let L_2 be the line passing through the point P_2=(5, -4, 4) with the same direction vector. Find the shortest distance d between these two lines, and find a point Q1 on L^and a point Q_2 on L_2 so that d(Q_1,Q_2) = d. Use the squareroot symbol \'v where needed to give an exact value for your answer. d= 0 Q1 = (0, 0, 0) Q2 = (o, o, 0)

Solution

given P1(3,-3,-3) with direction d[-2,-1,1]^T, P2(5,-4,4)

L1 = <3-2t, -3-t, -3+t>

d = distance from P2 to point <3-2t, -3-t, -3+t> on line L1

d² = (5-3+2t)² + (-4+3+t)² + (4+3-t)²

    = (2+2t)² + (-1+t)² + (7-t)²

    = 6t² - 8t + 54

   by solving the above equation we get t = 2/3

then d =(51.33)^1/2

here d>2 Since shortest distance between the two lines > 2, there are no points Q1 on L1 and Q2 on L2 such that d(Q1, Q2) = 2