An object of mass 10 moves with position function $r(t) = < 1+3\cos(t), 4\cos(t), −1+5\sin(t) >$ Find the force vector acting on this object at time t. Show that it points from the object towards the point $(1, 0, −1)$.
I believe that I have found the correct force vector: $F(t) = 10<-3\cos t,-4\cos t,-5\sin t>$
However, I am not sure how to figure out how to show that it points from the object towards the point $(1,0,-1)$.
Any help would be greatly appreciated.
What you need to do is describe the line along $\vec F$ at time $t$. You know that it is going through $\vec r(t)$. So the equation of this line is $$\vec l_\alpha=\vec r(t)+\alpha\vec F(t),\ \alpha\in\mathbb R$$ Plugging in what you calculated so far, $$\vec l_\alpha(t)=<1+3\cos t-30\alpha \cos t, 4\cos t-40\alpha\cos t, -1+5\sin t-50\alpha\sin t>$$ Now let's calculate the intersection by saying that $\vec l_\alpha(t_1)=\vec l_\beta(t_2)$. Then $$1+3(1-10\alpha)\cos t_1=1+3(1-10\beta)\cos t_2\\4(1-10\alpha)\cos t_1=4(1-10\beta)\cos t_2\\-1+5(1-10\alpha)\sin t_1=-1+5(1-10\beta)\sin t_2$$ If we want this intersection to be time independent, we see that $\alpha=\beta=0.1$ Plugging in this value of $\alpha$, you get a single point $<1,0,-1>$.