For a homework assignment, one of the questions requires finding the points where a parametric line in vector form intersects with a sphere whose radius changes. The line $r(s)$ is denoted by $\textbf{OP} + s\textbf{v}$, where $P = (x,y,z)$, and $\textbf{v} = \langle a,b,c\rangle$ (In the problem, there are actual values for the variables, but I'd like to look at it as a more general solution to help actually understand the problem). The sphere has a radius $t$, which grows as $t$ increases. The part of the problem that I'm having trouble understanding is how to find the points of intersection between the sphere and $r(s)$. I know that all intersections will fall on the line, but I'd like to find which value of $t$ will result in the first intersection, when $r(s)$ is tangent to the sphere's surface. When I visualize a similar problem in $\mathbb{R}^2$, it becomes easier to see when a circle intersects a line, but I have trouble finding out how to apply this in $\mathbb{R}^3$, and with a line that is parametric. Any nudges in the right direction would greatly help. Login To add answer/comment