10. AC

Solution: $\left| \overrightarrow { O P _ { 1 } } \right| = \left| \overrightarrow { O P _ { 2 } } \right| = 1$, so A is correct; $\left| \overrightarrow { A P _ { 1 } } \right| ^ { 2 } = 2 - 2 \cos \alpha , \left| \overrightarrow { A P _ { 2 } } \right| ^ { 2 } = 2 - 2 \cos \beta$, so B is incorrect; $\overrightarrow { O P _ { 1 } } \cdot \overrightarrow { O P _ { 2 } } \sin \alpha \sin \beta = \cos ( \alpha + \beta ) = \overrightarrow { O A } \cdot \overrightarrow { O P _ { 3 } }$, so C is correct; $\overrightarrow { O P _ { 2 } } \cdot \overrightarrow { O P _ { 3 } } = \cos ( \alpha + \beta ) \cos \beta - \sin ( \alpha + \beta ) \sin \beta \neq \overrightarrow { O P _ { 1 } } = \cos \alpha$, so D is incorrect. The answer is $AC$.