Answer
$$\operatorname{proj}_{{v}} {u} =\frac{-18}{26}(1,-5).$$
Work Step by Step
Let $u=(2,4)$, $v=(1,-5)$, $\langle{u}, {v}\rangle=u\cdot v$.
Then, we have $$\langle{u}, {v}\rangle=-18, \quad \langle{v}, {v}\rangle=26.$$ Now, the orthogonal projection of $u$ onto $v$ is given by $$\operatorname{proj}_{{v}} {u} =\frac{\langle{u}, {v}\rangle}{\langle{v}, {v}\rangle} {v}=\frac{-18}{26}(1,-5).$$
