I don't understand this reasoning from a solution.
Q. Let $S^n=\{(x_0,...,x_n) \in \mathbb{R}^{n+1}:x_0^2+...+x_n^2=1\}$ and $X=S^n-\{1,0,0,...,0\}$ and $Y=\{(y_0,...,y_n):y_0=0\}$. Find $\mu$ such that $g:Y \to X$ as a map and $g(y)$ is a unique point in $X$ so that $g(y)=\mu\{1,0,..,0)\}+(1-\mu)y$ is satisfied.
Now, I get the solution up to the following point
$||g(y)||^2=1$ since $g(y) \in X$. Therefore we obtain a quadratic equation in terms of $\mu$. (me; so we solve for $\mu$ right?)
However, it says
We need not solve the quadratic using the quadratic formula since we know that one solution is $\mu=1$.
Uhm, huh? How? where did $\mu=1$ come from? How do we "know"? Needs more explanation, it seems like there's a leap in logic here can someone see why? Please explain!
When $\mu=1$ we have the point is just $(1,0,0,\ldots, 0)$ by considering how the geometry works. Recall that the line equation and the sphere equation are two separate things, and their intersection, when both the line equation holds AND the sphere equation holds is just two points. But when both equations hold is exactly when you are allowed to substitute one into the other, which is what you are doing when you write $\lVert g(y)\rVert^2=1$. So you know that the solution set to this will be the two points where the line intersects the sphere. But by definition of the line, you know that $(1,0,0,\ldots , 0)$ is in the intersection, and looking back at $g(y)$ you see that happens when $\mu=1$.