An object of mass 10 moves with position function r ( t ) =&lt; 1 + 3 cos ⁡ ( t ) , 4 cos ⁡ ( t ) , − 1 + 5 sin ⁡ ( t ) &gt; 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 &lt; − 3 cos ⁡ t , − 4 cos ⁡ t , − 5 sin ⁡ t &gt;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.

Question

An object of mass 10 moves with position function   r  (  t  )  =&amp;lt;  1  +  3  cos  ⁡  (  t  )  ,  4  cos  ⁡  (  t  )  ,  −  1  +  5  sin  ⁡  (  t  )  &amp;gt; 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  &amp;lt;  −  3  cos  ⁡  t  ,  −  4  cos  ⁡  t  ,  −  5  sin  ⁡  t  &amp;gt;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.

na1p1a2pafr · Accepted Answer

What you need to do is describe the line along             F      →        at time t. You know that it is going through             r      →        (  t  ). So the equation of this line is                    l        →              α    =            r      →        (  t  )  +  α            F      →        (  t  )  ,     α  ∈      R  Plugging in what you calculated so far,                    l        →              α    (  t  )  =&amp;lt;  1  +  3  cos  ⁡  t  −  30  α  cos  ⁡  t  ,  4  cos  ⁡  t  −  40  α  cos  ⁡  t  ,  −  1  +  5  sin  ⁡  t  −  50  α  sin  ⁡  t  &amp;gt;Now let&#039;s calculate the intersection by saying that                     l        →              α    (      t    1    )  =                    l        →              β    (      t    2    ). Then  1  +  3  (  1  −  10  α  )  cos  ⁡      t    1    =  1  +  3  (  1  −  10  β  )  cos  ⁡      t    2      4  (  1  −  10  α  )  cos  ⁡      t    1    =  4  (  1  −  10  β  )  cos  ⁡      t    2      −  1  +  5  (  1  −  10  α  )  sin  ⁡      t    1    =  −  1  +  5  (  1  −  10  β  )  sin  ⁡      t    2  If we want this intersection to be time independent, we see that   α  =  β  =  0.1 Plugging in this value of   α, you get a single point   &amp;lt;  1  ,  0  ,  −  1  &amp;gt;

An object of mass 10 moves with position function r(t)=<1+3cos(t),4cos(t),−1+5sin(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).

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