${\displaystyle {\vec {F}}\cdot d{\vec {\ell }}=\left(F_{x}{\hat {i}}+F_{y}{\hat {j}}\right)\cdot \left({\hat {i}}dx+{\hat {j}}dy\right)}$ ${\displaystyle =F_{x}dx({\hat {i}}\cdot {\hat {i}})+F_{y}dy({\hat {j}}\cdot {\hat {j}})+(junk)({\hat {j}}\cdot {\hat {i}})}$

where ${\displaystyle {\hat {i}}\cdot {\hat {i}}={\hat {j}}\cdot {\hat {j}}=1}$ and ${\displaystyle {\hat {i}}\cdot {\hat {j}}=0}$.

For polar coordinates, use ${\displaystyle d{\vec {\ell }}={\hat {r}}dr+rd\theta {\hat {\theta }}}$