# Suppose we have two non-parallel vectors in R^3. Now, if we were to randomly select another vector in R^3, what is the probability that that new vector lies in the span of the first two vectors?

Geometric probability (with regards to span)
Suppose we have two non-parallel vectors in ${\mathbb{R}}^{3}$.
Now, if we were to randomly select another vector in ${\mathbb{R}}^{3}$, what is the probability that that new vector lies in the span of the first two vectors?
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smh3402en
Step 1
One way to model this, so everything becomes finite is to work on the unit sphere ${S}^{2}$. We may assume that the hyperplane is $H=\left\{z=0\right\}\subset {\mathbb{R}}^{3}$.
Step 2
The probability that a random vector of ${S}^{2}$ doesn't lie on the great circle $H\cap {S}^{2}$ would be exactly $\frac{|{S}^{2}\setminus \left(H\cap {S}^{2}\right)|}{|{S}^{2}|}=1.$.
The only problem here is that we exclude the zero vector, but I assume that this is not a big deal.