Question

Hello, I have a "graphical" problem. Let's say you have two points, 1 and 2, with some initial position. Let's make the initial position vectors r10 and r20. Let the points have constant velocity vectors v1 and v2. This gives, as a function of time,
r1(t) = r10 + v1*t
r2(t) = r20 + v2*t

The problem is to find at what time t the line segment between the two points intersects the origin. I had two approaches to this problem - The first, I made a line between the two points which changes with time and inserted x = 0 and y = 0 into the line's equation.

This gave me another equation. In addition to the four equations from the vectors (2 for each from x and y), I have 5 equations, and 5 unknowns (r1, r2, and t - r1 and r2 have two unknowns each). Solving for t using Mathematica gave me one very long answer. So that solution I'm confident in. Then, I tried a different approach. I made a conjecture that if r1(t)/|r1(t)| =-r2(t)/|r2(t)| then the two position vectors must be parallel and therefore intersect the origin with the line segment. Sounds good, right? However, solving this one equation for t gave a completely different answer than that of the previous solution... And there's where I'm stuck.

@telliott99 No I need to solve symbolically.

Answer 1

Do you have actual values? Maybe a diagram will help...

Answer 2

The attached example1.pdf gives the initial condition where
r1(0) = r10 = {1,1} and r2(0) = r20 = {-2,-3}

The line between them, as shown, does not intersect with the origin. Also, shown, however, are velocity vectors
v1 = {1,1} and v2 = {0, 1}.

Thus you can guess that at t = 1, you'll have
r1(1) = {2,2} and r2(1) = {-2,-2}.

That, of course, goes through the origin. The answer for this case would be t = 1. But I'm looking for a symbolic answer.

Answer 3

Sorry, but I gotta go. Great problem.

Answer 4

It's cool, right? Not a homework problem like the other poor summer school kids here...

Answer 5

Minor note about my "example" - the vector for the point {-2, -3} is off. It goes from {-2, -3} to {0,1} but it should go from {-2, -3} to {-2, -2}. Sorry.

Answer 6

i guess equating x,y,z coordinates and solving for t would help to find the value of t.

Answer 7

Only 2D

Answer 8

\[ x_1 = x_2 \implies x_{10} + t{v_x1} = x_{20} + tv_{x2} \]
this will give us
\[ t = {x_1 - x_2 \over v_{x2} - v_{x1}}\]
if we put these t's in y's, both y's must have same values ... also same goes for z.
that will help us to determine some unknown parameters.
or if values are given for x , y, z component of velocities, it can be determined if they intersect or not based on these conditions

Answer 9

As I said, 2D only. I don't know why you're insisting with the z's. Also, I already tried solving "brute force" with mathematica. It worked, and I have a huge solution. The problem is that the huge solution is not equal to the solution I get from my second approach.

Answer 10

2d cases only ... it's a bit simpler. do you have any other constraints for v?

Answer 11

It's a constant. I can have my solution in terms of the initial positions and the initial velocities.

Answer 12

the condition is

if you simplify this ... you should get the required condition.