Let O be the origin, and M and N be the points on the lines (x-5)/4=(y-4)/1=(z-5)/3 and (x+8)/12=(y+2)/5=(z+11)/9 respectively such that MN is the shortest distance between the given lines. Then O M⃗ · O N⃗ is equal to _________.

Solution

Maths sir · 4 steps

Write M and N with one parameter each, then make MN perpendicular to both lines.

1. General points. On the first line take M = (5+4s, 4+s, 5+3s). On the second line take N = (-8+12t, -2+5t, -11+9t). Then MN⃗ = (-13+12t-4s, -6+5t-s, -16+9t-3s).
2. Perpendicular to both. The shortest-distance segment is perpendicular to both direction vectors (4,1,3) and (12,5,9). Set each dot product with MN⃗ to zero. After simplifying you get two linear equations.

   80t - 26s = 106 ⇒ 40t - 13s = 53 ; 250t - 80s = 330 ⇒ 25t - 8s = 33

3. Solve. Multiply the first equation by 8 and the second by 13 and subtract. This gives 5t = 5, so t = 1 and then s = -1. So M = (1, 3, 2) and N = (4, 3, -2). Check: MN⃗ = (3, 0, -4) is perpendicular to both directions.
4. So. OM⃗ · ON⃗ = 1 × 4 + 3 × 3 + 2 × (-2) = 4 + 9 - 4 = 9.

The idea to keep. To find the actual feet of the common perpendicular, use one parameter per line and make the joining vector perpendicular to both directions. That gives two equations in two unknowns.

Answer from the NTA final answer key. Solution by PrepWiser.