Why don't you look at a text that descrives the formulau.v= |u||v|cos θ to understand what the relationship is between vectors u, v and angle θ? You can consult your textbook or Wikipedia here and here. Again, what does the text say concerning how θ is defined by the two vectors? Can you say that angle OZY, which you need to find, is defined by vectors OZ and OY, or by OZ and ZY, in this way?

If $CP:PZ = \lambda$, then

\[

OP=\frac{OC+\lambda OZ}{\lambda+1}

\]

It may be easier to remember this formula in a more general case, where $CP:PZ = \lambda:\mu$; then

\[

OP=\frac{\mu OC+\lambda OZ}{\lambda+\mu}

\]

Note that $OC$ and $OZ$ are multiplied by $\lambda$ and $mu$ in a criss-cross manner. Indeed, $\lambda$ corresponds to the initial segment $CP$ and $\mu$ corresponds to the final segment $PZ$, yet the radius-vector $OC$ of the initial point $C$ is multiplied by $\mu$, while the radius-vector $OZ$ of the final point $Z$ is multiplied by $\lambda$. This is justified by the limit cases: when $\lambda=0$, all contribution is done by vector $OC$ and $P=C$; when $\mu=0$, all contribution is done by $OZ$ and $P=Z$. Of course, $OC$, $OZ$ and so on above are vectors, not just segment lengths.

Use the fact that $OP\cdot CZ=0$.