Exercise 3 Page 381

For two rays to be the same, they must have the same endpoint and point in the same direction. Let's look at each of the pairs of rays one at a time and see if they meet these conditions.

$\overrightarrow{KP}$ and $\overrightarrow{PK}$

The ray $\overrightarrow{KP}$ has the endpoint $K,$ goes through the point $P,$ and continues indefinitely. Note that point $P$ is the intersection between the two lines $\overleftrightarrow{KL}$ and $\overleftrightarrow{MN}.$ Let's highlight $\overrightarrow{KP}$ and $P$ in the diagram to make it easier for us to visualize the ray.

What about the ray $\overrightarrow{PK}?$ It has the endpoint $P,$ goes through the point $K,$ and continues on indefinitely. Since they have different endpoints and point in different directions, we can draw the conclusion that this ray is not the same as $\overrightarrow{KP}.$ We can highlighted $\overrightarrow{PK}$ in the diagram to make the differences more apparent.

$\overrightarrow{NP}$ and $\overrightarrow{NM}$

$\overrightarrow{NP}$ has the endpoint $N,$ goes through $P,$ and continues indefinitely. Let's highlight this ray in the diagram.

$\overrightarrow{NM}$ also has the endpoint $N.$ Remember, the two rays are only the same if they also point in the same direction. $\overrightarrow{NM}$ must travel through point $P$ on its way to go through the point $M.$ This can be seen on the diagram above. We can know for sure that this is the case since the points $N,$ $P,$ and $M$ are all collinear. This means that the two rays are indeed pointing in the same direction, and are, therefore, the same.