Geometric probability (with regards to span)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?

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Geometric probability (with regards to span)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?

smh3402en · Accepted Answer

Step 1One 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  =  {  z  =  0  }  ⊂            R        3  .Step 2The probability that a random vector of       S    2   doesn&#039;t lie on the great circle   H  ∩      S    2   would be exactly                     |                    S        2            ∖      (      H      ∩              S        2            )              |                            |                    S        2                    |              =  1..The only problem here is that we exclude the zero vector, but I assume that this is not a big deal.

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?

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