1. The problem statement, all variables and given/known data Let L1 be the line (0,4,5) + (1,2,-1)t. Let L2 be the line (-10,9,17) + (-11,3,1)t. Find the line L passing through and orthogonal to L1 and L2. What is the distance between L1 and L2? 2. Relevant equations Vector Projection Equation: V • W/|W| 3. The attempt at a solution I think finding the equation is the more difficult part of the question, so I'll begin by finding the distance. First, I find a vector orthogonal to both lines by cross product. This vector is <5,10,25>, which I reduce to <1,2,5>. Using the points given in the definitions of lines, I subtract to get another vector, <10,-5,22>. By the equation for vector projections, <10,-5,22> • <1,2,5> = 110, which I divide by the magnitude of <1,2,5>. So, the distance equals 110/sqrt(30). Now, I want the equation for this orthogonal line to be in the form (a,b,c) + <1,2,5>*s, where (a,b,c) is a point on the line. By multiplying out, xs: a + s ys: b +2s zs: c + 5s I suppose I want to solve for s in terms of t. I know xt1: t yt1: 4 + 2t zt1: 5 - t So, t = a + s, then.... I think this whole system of equations will turn into a large mess. Is there a simpler way to approach this problem?