This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To find the shortest distance between the two skew lines L and M, we use the formula: D = |(P_L P_M) · (d_L × d_M)|\|d_L × d_M\| where d_L and d_M are the direction vectors of lines L and M, respectively, and P_L P_M is a vector connecting a point P_L on line L to a point P_M on line M. From the given equations: Line L: (x+8)/(2) = (y-10)/(3) = (z-6)/(1) Line M: (x-1)/(-1) = (y-1)/(2) = (z-1)/(4) Step 1: Identify direction vectors and points. For line L: d_L = 2 \\ 3 \\ 1 and P_L = (-8, 10, 6). For line M: d_M = -1 \\ 2 \\ 4 and P_M = (1, 1, 1). Step 2: Calculate the vector P_L P_M. P_L P_M = P_M - P_L = (1 - (-8), 1 - 10, 1 - 6) = (9, -9, -5) Step 3: Calculate the cross product of the direction vectors, d_L × d_M. d_L × d_M = i & j & k \\ 2 & 3 & 1 \\ -1 & 2 & 4 = i((3)(4) - (1)(2)) - j((2)(4) - (1)(-1)) + k((2)(2) - (3)(-1)) = i(12 - 2) - j(8 - (-1)) + k(4 - (-3)) = 10i - 9j + 7k So, d_L × d_M = 10 \\ -9 \\ 7 . Step 4: Calculate the magnitude of the cross product, \|d_L × d_M\|. \|d_L × d_M\| = sqrt(10^2 + (-9)^2 + 7^2) = sqrt(100 + 81 + 49) = sqrt(230) Step 5: Calculate the scalar triple product, (P_L P_M) · (d_L × d_M). (P_L P_M) · (d_L × d_M) = (9)(10) + (-9)(-9) + (-5)(7) = 90 + 81 - 35 = 171 - 35 = 136 Step 6: Calculate the shortest distance. D = (|136|)/(sqrt(230)) D = (136)/(sqrt(230)) To rationalize the denominator, multiply the numerator and denominator by sqrt(230): D = 136sqrt(230)230 Simplify the fraction: D = 68sqrt(230)115 The shortest distance between the lines L and M is 68sqrt(230)115. That's 2 down. 3 left today — send the next one.