Line segment multiplication

Question

I was just reviewing the inverse curve and there is a condition, the two lines $OP$ and $OQ$ should multiply to the radius $k$ squared: $OP\cdot OQ = k^{2}$. By calculus I found the magnitude of each line from the origin to these antipodal points on a circle through their dot products: $\sqrt{OP \cdot OP}$ and $\sqrt{OQ \cdot OQ}$ with $O = (0,0)$, $P=(0,1)$ and $Q=(0,-1)$... they multiply to $1$. Initially though I tried finding the lines' lengths by computing $OP = 1-0=1$ and $OQ = -1 -0 = -1$; they didn't multiply to $1$. How come the length of of a line segment such as the interval $[2,2.5]$ is $2.5 - 2$ but the first computation I did isn't working?

Answer 1

Even though we write them the same way, the vector (0,1) and the interval (0,1) are very different things! In the case of the interval [2,2.5], 2 is the start of the interval and 2.5 is the end of the interval, so subtracting them gives the distance. In the case of the vector (0,1), this vector starts at the origin (0,0) on the 2d plane and ends at the point (0,1). Both of these numbers are ends: one in the x direction and one in the y direction. Because they're different directions, it usually doesn't make sense to add or subtract them.