10. Given vectors $\vec{a}, \vec{b}, \vec{c}, \vec{d}$, and at least one of $\vec{a}, \vec{b}$ is not a zero vector, the function of real number $t$ is $f(t)=|t \vec{a}+\vec{c}|^{2}+|t \vec{b}+\vec{d}|^{2}$, then when $t=$ $\qquad$, $f(t)$ reaches its minimum value.
10. $\begin{array}{l}-\frac{\vec{a} \cdot \vec{c}+\vec{b} \cdot \vec{d}}{|\vec{a}|^{2}+|\vec{b}|^{2}} \\ f(t)=t^{2} \vec{a}^{2}+2(\vec{a} \cdot \vec{c}) t+\vec{c}^{2}+t^{2} \vec{b}^{2}+2(\vec{b} \cdot \vec{d}) t+\vec{d}^{2} \\ =\left(\vec{a}^{2}+\vec{b}^{2}\right) t^{2}+2(\vec{a} \cdot \vec{c}+\vec{b} \cdot \vec{d}) t+\vec{c}^{2}+\vec{d}^{2} .\end{array}$