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?

Jensen Mclean 2022-10-06 Answered
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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Answers (1)

smh3402en
Answered 2022-10-07 Author has 11 answers
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 = { z = 0 } R 3 .
Step 2
The probability that a random vector of S 2 doesn'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.
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