Determine if the following lines are parallel, intersecting, or skew: L: (x+8)/2 = (y-10)/3 = (z-6)/1 and M: (x-1)/-1 = (y-1)/2 = (z-1)/4

Welcome back — been a while! Let's pick up where you left off. To determine the relationship between the two lines, we first extract their direction vectors and a point on each line. The symmetric equations of the lines are given as: 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, the direction vector is d_L = 2 \\ 3 \\ 1 and a point on the line is P_L = (-8, 10, 6). For line M, the direction vector is d_M = -1 \\ 2 \\ 4 and a point on the line is P_M = (1, 1, 1). Step 2: Check if the lines are parallel. Lines are parallel if their direction vectors are scalar multiples of each other. We check if d_L = k d_M for some scalar k. 2 \\ 3 \\ 1 = k -1 \\ 2 \\ 4 From the x-components: 2 = -k k = -2. From the y-components: 3 = 2k k = (3)/(2). Since the values of k are not consistent (-2 ≠ (3)/(2)), the direction vectors are not parallel. Therefore, the lines are not parallel. Step 3: Check if the lines intersect. To check for intersection, we write the parametric equations for each line and set the corresponding coordinates equal. Parametric equations for L: x_L = -8 + 2t y_L = 10 + 3t z_L = 6 + t Parametric equations for M: x_M = 1 - s y_M = 1 + 2s z_M = 1 + 4s Set the coordinates equal: 1) -8 + 2t = 1 - s 2) 10 + 3t = 1 + 2s 3) 6 + t = 1 + 4s From equation (3), express t in terms of s: t = 1 + 4s - 6 t = 4s - 5 Substitute this expression for t into equation (1): -8 + 2(4s - 5) = 1 - s -8 + 8s - 10 = 1 - s 8s - 18 = 1 - s 9s = 19 s = (19)/(9) Now substitute s = (19)/(9) back into the expression for t: t = 4((19)/(9)) - 5 t = (76)/(9) - (45)/(9) t = (31)/(9) Finally, check if these values of s and t satisfy equation (2): 10 + 3t = 1 + 2s 10 + 3((31)/(9)) = 1 + 2((19)/(9)) 10 + (31)/(3) = 1 + (38)/(9) (30)/(3) + (31)/(3) = (9)/(9) + (38)/(9) (61)/(3) = (47)/(9) This statement is false, as (61)/(3) = (183)/(9). Since the values of s and t do not satisfy all three equations, the lines do not intersect. Step 4: Determine the relationship. Since the lines are not parallel and do not intersect, they are skew. We can also check if they are perpendicular by calculating the dot product of their direction vectors: d_L · d_M = (2)(-1) + (3)(2) + (1)(4) = -2 + 6 + 4 = 8. Since the dot product is not zero, the lines are not perpendicular. The relationship between lines L and M is that they are skew lines. The final answer is Skew lines Send me the next one 📸