Does relative velocity vector lie on the line?

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Cassy A
Cassy A 2017-6-19
编辑： Cassy A 2017-6-20
Hello everyone! I am new to matlab and would like some assistance. For two particles A and B with given position and constant velocity vectors, I would like to show in matlab whether their relative velocity vector (V_A - V_B) lies on the line joining them.

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回答（2 个）

Julian Hapke
Julian Hapke 2017-6-19
编辑：Julian Hapke 2017-6-19
define a vector between the two points and check if the cross product of the delta velocity and the connection vector is 0. so if
ra = [xa,ya,za]
rb = [xb,yb,zb]
% va and vb being the velocities
test = cross(ra-rb,va-vb)
if ~test % check if all are zero
disp('same direction')
end
EDIT: As Jan Simon pointed out, you may get precision problems, so
if all(abs(test))<eps % or any other threshold
disp('same direction')
end
Jan
Jan 2017-6-19
编辑：Jan 2017-6-19
Do you want to calculate the angle between the connection of the 2 points and the relative velocity? Then:
u = A - B;
v = vA - vB;
a = atan2(norm(cross(u,v)), dot(u,v));
Now check if the result a is below a certain limit. You cannot expect it to be exactly 0.0 or 180.0 due to the limited precision of the floating point values. Perhaps this is a smart limit:
limit = 10 * eps(max([A(:); B(:); vA(:); vB(:); u(:); v(:)])
isParallel = abs(a) < limit || abs(a) - 180 < limit;
But there is not "best" definition of the limit.
See https://www.mathworks.com/matlabcentral/answers/101590-how-can-i-determine-the-angle-between-two-vectors-in-matlab for a discussion, why atan2 is more accurare than acos or asin or the corresponding cross and dotr product methods only.
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Les Beckham
Les Beckham 2017-6-20
编辑：Les Beckham 2017-6-20
I have to say that, if you created the plots in your posts using Matlab, then I am very impressed with your Matlab graphics skills (especially the second 3D one). If not, I would be curious about how you did create them.
It appears that you are dealing with a problem that, in missile guidance terms, would be referred to as 'line-of-sight' guidance. The basic idea is that if the 'line-of-sight' vector (the apparent position of B as seen by A) stays constant then the two 'particles' are on a collision course. You may want to search for 'line of sight (or 'LOS') guidance' for more information. It is actually regulating the relative velocity of one object as 'seen' by the other (LOS rate) to zero that will result in a collision course.
Jan and Julian have provided good answers that should point you in the right direction on the Matlab coding side.
Cassy A
Cassy A 2017-6-20
编辑：Cassy A 2017-6-20
Hey! I used a program called GeoGebra. It's fun to play with :)
Thank you for all the help though. You guys are great! :)

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