Question

Two ants, red ant and black ant, creeping along lines in 3-space. At time t, red ant is at point on the line
x=3-t , y-2=2t , z-2t=6 and at the same time t, black ant is at the point on the line
2x=t , y-3t=2 , z=2+t a) Find the starting distance between red and black ants before they travel.
b) Is that possible red ant and black ant will collide each other? If yes, when and where will the collision happen?

Answer 1

a. The distance between the red ant and the black ant is five units.

To represent the red ant,
xr = 3 - t 
yr = 2 + 2 t 
zr = 6 + 2 t 

To represent the black ant, 
xb = t/2 
yb = 2 + 3 t 
zb = 2 + t 

To calculate,
at t = 0 
xb - xr = -3 
yb - yr - 2-2 = 0 
zb - zr = 2-6 = -4 
Therefore,
d = √(9+16) = √(25) = 5

b. There will be no collision between the two ants.

To calculate,
3 - t = t/2
6 - 2t = t
6 = 3t
t = 2 

If t = 2,
2 + 2*2 = 2 + 3*2