# Find sphere intersection point

I have a sphere which is originated at:

$$\begin{matrix} 0 & 0 & 0 \end{matrix}$$

Its radius r is 150. I have a line which goes from: $$\begin{matrix} 0 & 0 & 0 \end{matrix}$$

and passes through the point inside a sphere [x y z]. Knowing that line starts from origin and crosses [x y z], how to find out at what coordinates line will intersect with the sphere?

The line consists of the points $$[\lambda x,\lambda y,\lambda z]$$ with $$\lambda\in\Bbb{R}$$. The sphere consists of the points $$[u,v,w]$$ with $$u^2+v^2+w^2=150^2.$$ So the points of intersection are the points $$[\lambda x,\lambda y,\lambda z]$$ with $$\lambda\in\Bbb{R}$$ such that $$(\lambda x)^2+(\lambda y)^2+(\lambda z)^2=150^2.$$ It is not hard to see that the appropriate values for $$\lambda$$ are $$\lambda=\pm\sqrt{\frac{150^2}{x^2+y^2+z^2}}=\pm\frac{150}{\sqrt{x^2+y^2+z^2}}.$$