# Translations:The unit tangent vector/8/en

Let us review the results. Let the vector r be a point on the raypath. A closely spaced point on the raypath can be represented by the vector ${\displaystyle \mathbf {r} +d\mathbf {r} }$. Their difference is the vector ${\displaystyle d\mathbf {r} }$, which is the vector that connects the two points in question. Let the path length between the two points be ds. The unit tangent to the raypath is the limit ${\displaystyle d\mathbf {r} /ds}$ as the points approach each other. The length of the vector ${\displaystyle d\mathbf {r} }$ is approximately equal to the path-length difference ds. As a result, the vector ${\displaystyle d\mathbf {r} /ds}$ is the unit vector. Let ${\displaystyle \theta }$ be the angle that the tangent makes with the horizontal axis. The unit tangent vector is then ${\displaystyle \mathbf {u=} \left({\rm {\ cos\ }}\theta {\rm {,\ \ sin\ }}\theta \right)}$. The vector u is directed along the tangent to the curve in the direction of increasing values of the arc length s.