Write $a$ in the form $a=a_{T} T+a_{N} N$ at the given value of $t$ without finding $T$ and $N$. \[ r(t)=(3 t+3) i+(2 t) j+\left(3 t^{2}\right) k, \quad t=-1 \] \[ a= \] \[ T+(\square N \] (Type exact answers, using radicals as needed.)

Step 1 ：Given the position vector \(r(t)=(3t+3)i+(2t)j+(3t^2)k\), we first find the velocity vector \(v\) which is the first derivative of the position vector \(r(t)\).

Step 2 ：The velocity vector \(v\) is \(v=3i+2j+6t k\).

Step 3 ：Next, we find the acceleration vector \(a\) which is the derivative of the velocity vector \(v\).

Step 4 ：The acceleration vector \(a\) is \(a=6k\).

Step 5 ：The acceleration vector \(a\) at \(t=-1\) is a constant vector with components \((0, 0, 6)\).

Step 6 ：However, without knowing the unit tangent vector \(T\) and the unit normal vector \(N\), we cannot express \(a\) in the form \(a=a_{T} T+a_{N} N\).

Step 7 ：Final Answer: The acceleration vector \(a\) at \(t=-1\) is \(\boxed{(0, 0, 6)}\). However, we cannot express \(a\) in the form \(a=a_{T} T+a_{N} N\) without knowing \(T\) and \(N\).