A linear map f:RR^3 to RR^3 is defined as vec(u) xx (vec(r) xx vec(u)) where vec(u) is a unit vector. What would its geometry look like?

Javion Henry 2022-07-21 Answered
A linear map f : R 3 R 3 is defined as u × ( r × u ) where u is a unit vector. What would its geometry look like?
I know that I can rewrite this map as ( u u ) r ( u r ) u = r ( u r ) u
However I am not sure what to do from here.
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Answers (1)

salumeqi
Answered 2022-07-22 Author has 15 answers
Clearly ( u r ) u is the component of r along the unit vector u . So u × ( r × u ) = r ( u r ) u is the component of r which is perpendicular to the unit vector u .
The map is neither injective nor surjective. It is not injective because for a and for a + λ u we get the same value for any real λ. It is not surjective because the range of the function doesn't contain any vector in R 3 that has non zero component in the direction of u such as u itself.

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