Problem on writing down a given vector field

Write down the vector field $\mathbf{F}(x,y)$ whose value at $(x,y)$ is of unit length and points toward $(1,0)$.
• Solution

To specify the value of $\mathbf{F}(x,y)$, we first construct a vector that goes from$(x,y)$ to $(1,0)$. Then, we will modify it so that it has unit length.
Recall that
At $(x,y)$, a vector pointing toward $(1,0)$ is $\langle 1-x, -y \rangle$.
Recall that
Thus, we are after a vector of length 1 in the direction of $\langle 1-x, -y \rangle$.
This unit vector is $$\text{dir } \langle 1-x, -y \rangle = \frac{\langle 1-x, -y \rangle}{\left | \langle 1-x, -y \rangle \right |}$$
At each point $(x,y)$, a unit vector in the direction toward $(1,0)$ is $$\mathbf{F}(x,y) = \frac{\langle 1-x, -y \rangle}{\sqrt{(1-x)^2 + y^2}}$$
We comment that this formula does not make sense at $(1,0)$, which is ok because there is no direction "toward $(1,0)$" at that point.