1. Given position vector

   $\vec r(t)=10t\,\hat i+15t^2\,\hat j+7\,\hat k$

2. Find velocity by differentiating position with respect to time:

   $\vec v=\frac{d\vec r}{dt}=10\,\hat i+30t\,\hat j+0\,\hat k$

3. Find acceleration by differentiating velocity with respect to time:

   $\vec a=\frac{d\vec v}{dt}=0\,\hat i+30\,\hat j+0\,\hat k$

4. Relate force and acceleration using Newton's second law:

   $\vec F=m\vec a$

   Since mass $m$ is positive, the direction of force is the same as the direction of acceleration.

5. Direction of acceleration:

   $\vec a=30\,\hat j$

   This is along the positive $y$-axis.

6. Check options:

   • A: Positive $x$-axis $\rightarrow$ Incorrect
   • B: Positive $y$-axis $\rightarrow$ Correct
   • C: Positive $z$-axis $\rightarrow$ Incorrect
   • D: In $x$-$y$ plane $\rightarrow$ Not the most specific correct direction, since force is purely along $+y$

Therefore, the net force is along the positive $y$-axis.