I have two objects with position vectors $r_1(t)$ and $r_2(t)$, where $r_1(t)=(-5+16t)i+(-9+12t)j+(2+19t)k$ and $r_2(t)=(10-11t)i+(-12+15t)j+(1+20t)k$, and am trying to determine the closest approach.
When using a vector calculus method, the closest approach comes out at approximately 1.396 units, but when using scalar projection, the closest approach is approximately 0.8330 units.
I think the scalar projection method, for this case, is incorrect because it finds the magnitude of a vector that exists between a point on the path of one object and a point on the path of the other object and is orthogonal to the direction of each path, but does not account for how these points must exist for the same value of time.
Is this thought correct, if not, why is there a difference in the closest approach calculated with vector calculus and the closest approach calculated with scalar projection?