At time t=0 a particle is located at the point (1,2,3). it travels in a straight line to the point (4,1,4), has speed 2 at (1,2,3) and constant acceleration 3i-j+k. find an equation for the position vector r(t) of the particle at time t.

At time , the particle has position , so



You also know that at this point, when , the particle's speed is , which is the magnitude of the velocity at this point because the particle is traveling in one direction. This means



If is the time at which the particle arrives at the point , i.e. , then you have the system



Eliminating the terms with , you're left with the system



so you're left with solving this system with the constraint that the sum of these constants' squares is .

However, there are two solutions to this, with

Obviously, we require that , so we need to check if either option forces using an invalid value of . This amounts to finding such that . Indeed, this has two solutions, , or and . We ignore the first.

Plugging in this value of into the system , we find that

Therefore the equation for the position vector is