Linear Algebra with Reflection Across Axis

Question

Point $P(-2,5)$ is reflected on the line $y=\frac{2}{5}x$. How do you find the coordinates of the reflection using linear algebra and not the distance formula?

Answer 1

The vector representing the slope of the line is $\overrightarrow{m}=(run,rise)=(5,2)$. The vector representing the point is $\overrightarrow{P}=(-2,5)$

Notice that $\overrightarrow{P}$ is perpendicular to $\overrightarrow{m}$, so the vector $\overrightarrow{P}$ gives us the distance from $P$ to the line.

In particular, reflection of the point P is given by $$-\overrightarrow{P}=(2,-5)$$

(draw the picture and you will see why this works)