7. Let point $O$ be the origin, and the coordinates of points $A$ and $B$ be $(a, 0)$ and $(0, a)$, respectively, where $a$ is a positive constant. Point $P$ lies on the line segment $AB$, and $\overrightarrow{A P}=t \cdot \overrightarrow{A B}(0 \leqslant t \leqslant 1)$. Then the maximum value of $\overrightarrow{O A} \cdot \overrightarrow{O P}$ is
7. $a^{2}$ Hint: According to the problem, we can set $\overrightarrow{O A}=(a, 0), \overrightarrow{O P}=(a-a t, t a)$, so $\quad \overrightarrow{O A} \cdot \overrightarrow{O P}=a^{2}(1-t) \leqslant a^{2}$, when $t=0$, the above expression takes the equality, hence the maximum value of $\overrightarrow{O A} \cdot \overrightarrow{O P}$ is $a^{2}$