From the image above, I know the vector OA, OC, and I know that |OB| = 1,5 (so the radius of the sphere is 1.5).
"O" is the center of the sphere and it's coordinates are (0,0,0). I am trying to find a formula so that I can compute the value of the vector OB. "B" it's a point on the sphere that represents the intersection between the vector AC and the sphere (it's positioned on the border of the sphere).
I found this article (https://en.m.wikipedia.org/wiki/Line%E2%80%93sphere_intersection), but I'm not sure if it's helpful in my case because I wasn't able to find a solution by using it.
Does anyone have any ideas on how I could approach this? Thank you in advance for your help.
$A, B, C$ are colinear, so $\vec{B}=c\vec{A}+(1-c)\vec{C}$. Then, since $B$ lies on the sphere $x^2+y^2+z^2=\frac{9}{4}$, we must solve the equation $$||c\vec{A}+(1-c)\vec{C}||=\frac{3}{2}$$ for real constants $c$.
For example, if $\vec{A}=\vec{i}$ and $\vec{B}=\vec{j}$ then $8c^2-8c-5=0$ and we have two solutions: $B_1=(\frac{2+\sqrt{14}}{4},\frac{2-\sqrt{14}}{4},0)$ and $B_2=(\frac{2-\sqrt{14}}{4},\frac{2+\sqrt{14}}{4},0)$.