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?

red:

xr = 3 - t
yr = 2 + 2 t
zr = 6 + 2 t

black:
xb = t/2
yb = 2 + 3 t
zb = 2 + t

at t = 0
xb - xr = -3
yb - yr - 2-2 = 0
zb - zr = 2-6 = -4
so
d = sqrt(9+16) = 5 answer part a)

can

3-t = t/2 ?
only if
1.5 t = 3
t = 2
now when t = 2
2 + 2*2 = 2 + 3*2 whoops, no way

To find the starting distance between the red and black ants, we can calculate the distance between their starting points.

a) Starting point of the red ant:
At time t = 0, we substitute t = 0 into the equation of the red ant's line:
x = 3 - t = 3 - 0 = 3,
y - 2 = 2t = 2 * 0 = 0 → y = 2,
z - 2t = 6 → z - 2 * 0 = 6 → z = 6.

Therefore, the starting point of the red ant is (3, 2, 6).

Starting point of the black ant:
At time t = 0, we substitute t = 0 into the equation of the black ant's line:
2x = t = 0 → 2x = 0 → x = 0,
y - 3t = 2 - 3 * 0 = 2,
z = 2 + t = 2 + 0 = 2.

Therefore, the starting point of the black ant is (0, 2, 2).

Now, we can calculate the distance between the two points using the distance formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2).

Distance = √((3 - 0)^2 + (2 - 2)^2 + (6 - 2)^2)
= √(3^2 + 0^2 + 4^2)
= √(9 + 0 + 16)
= √(25)
= 5.

Therefore, the starting distance between the red and black ants is 5 units.

b) To determine if the red ant and black ant will collide, we need to find the time at which their paths intersect. We can set their corresponding x, y, and z coordinates equal to each other and solve for t.

For the x-coordinate:
3 - t = 2x = 2(0) = 0
3 - t = 0
t = 3.

For the y-coordinate:
y - 2 = 2t = 2(3) = 6
y - 2 = 6
y = 8.

For the z-coordinate:
z - 2t = 6 = 2 + t
z - 2t = 2 + t
z = 2t + 2 = 2(3) + 2 = 8.

Therefore, at time t = 3, both ants are at the point (0, 8, 8).

Hence, the collision between the red ant and black ant will happen at time t = 3 and at the location (0, 8, 8).