At time t=0, a particle is located at the point (3,6,9). it travels in a straight line to the point (5,2,7), has speed 8 at (3,6,9) and constant acceleration 2i−4j−2k. find an equation for the position vector of the particle.

The particle has constant acceleration according to

Its velocity at time is

Then the particle has position at time according to

At at the point (3, 6, 9), i.e. when , it has speed 8, so that

We know that at some time , the particle is at the point (5, 2, 7), which tells us

and in particular we see that

and

Then

That is, there are two possible initial velocities for which the particle can travel between (3, 6, 9) and (5, 2, 7) with the given acceleration vector and given that it starts with a speed of 8. Then there are two possible solutions for its position vector; one of them is

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