Given that,

$$\overrightarrow{\mathrm{P}}(t)=\mathrm{A}[\hat{i} \cos (k t)-\hat{j} \sin (k t)]$$

where, $$\mathrm{A}$$ and $$k=$$ constant

$$\begin{aligned}\text { Force }\overrightarrow{\mathrm{F}}) & =\frac{d \overrightarrow{\mathrm{P}}}{d t}\left(\text { from Newton's } 2^{\text {nd }}\right. \text { law) } \\\\ \Rightarrow \quad \overrightarrow{\mathrm{F}} & =\frac{d}{d t}[\mathrm{~A}(\hat{i} \cos (k t)-\hat{j} \sin (k t))] \\\\& =\mathrm{A} k[\hat{i} \cos (k t)-\hat{j} \sin (k t)]\end{aligned}$$

By applying dot product between $$\vec{F} \& \vec{P}$$ we have

$$\begin{aligned}& \overrightarrow{\mathrm{F}} \cdot \overrightarrow{\mathrm{P}}=0 \\\\ & \Rightarrow|\vec{F}||\vec{P}| \cos \theta=0 \\\\ & \Rightarrow \quad \cos \theta=0 \\\\ & \text {Hence,} \theta=90^{\circ}\end{aligned}$$