#### Answer

$$7$$

#### Work Step by Step

Since
$$\mathbf{r}=(1+2 t) \mathbf{i}+(1+3 t) \mathbf{j}+(6-6 t) \mathbf{k} $$
Then
\begin{align*}
\mathbf{v}&=2 \mathbf{i}+3 \mathbf{j}-6 \mathbf{k} \\
|\mathbf{v}|&=\sqrt{2^{2}+3^{2}+(-6)^{2}}=7
\end{align*}
Hence
\begin{align*}
s(t)&=\int_{0}^{t} 7 d \tau=7 t\\ \text{ Length }&=s(0)-s(-1)\\
&=0-(-7)=7
\end{align*}