Vector Math - Determine coordinations of bounding box There are two points on the 2D plane, $\langle 10, 10\rangle$ pointing to $\langle 30, 50\rangle$. How does one determine the coordinates of the blue line if it touches the bounding box of a $100\times100$ grid using Vector Math?

If you can provide a generalized solution (aka. formula) that would be awesome.
Since you haven't added any variables in your problem, I'm not sure what you want generalized, so I'll assume that the box is $[0,n]\times [0,n]$ and the ray starts at $(x_1,y_1)$ in the direction of $(x_2,y_2)$.

We can assume that $x_1\le x_2$ and $y_1\le y_2$; if not, you can reflect the picture horizontally and/or vertically to make it so. Then the ray will either hit the bottom or right side of the picture, and by swapping $x$ and $y$ coordinates (i.e. reflect along the top-left to bottom-right diagonal) we can make it always hit the bottom side of the picture. (But we don't know which side it will hit yet, so we will hold off on this for a bit.)

So we want to find the intersection of the line $((1-t)x_1+tx_2,(1-t)y_1+ty_2)$ for $t\in[0,\infty)$ with the line $y=n$, which happens exactly when $(1-t)y_1+ty_2=n$, or solving for $t$, this is $$y_1+t(y_2-y_1)=n\iff t(y_2-y_1)=n-y_1\iff t=\frac{n-y_1}{y_2-y_1},$$
and plugging in for $x$ we get that the intersection point is $((n-y_1)\frac{x_2-x_1}{y_2-y_1}+x_1,n)$.

But not so fast, recall that we still aren't sure if it will in fact hit the bottom side. The expression there will always touch the bottom line, but it may be outside the bounding box, if the first coordinate is greater than $n$. This happens when $$(n-y_1)\frac{x_2-x_1}{y_2-y_1}+x_1\ge n\iff \frac{x_2-x_1}{y_2-y_1}\ge\frac{n-x_1}{n-y_1},$$ and if this is true then the line is actually on track to hit the right side first, in which case you swap all the $x$'s and $y$'s to get $(n,(n-x_1)\frac{y_2-y_1}{x_2-x_1}+y_1)$ instead.
I'm not sure what you're trying to do, but a (n axis aligned) bounding box in 2D is typically just defined as the rectangle defined by the points (minX,minY) and (maxX, maxY), ie simply the minimum/maximum x and y coordinates of whatever you're considering. For your vector that'd be (10,10) and (30,50), meaning in your drawing that the upper-left corner of the box would be at (10,10) and the lower right corner at (30,50).

If you're really looking to find the intersection of your vector with that line of the rectangle you've drawn, you might want to look at https://stackoverflow.com/questions/14307158/how-do-you-check-for-intersection-between-a-line-segment-and-a-line-ray-emanatin. For your case your vector would be your ray and the line (0,100),(100,100) is the line segment.