I am not sure how I should explain the following problems:

If you take any two perpendicular unit vectors X and Y, then

1. X\[Cross]Y is perpendicular to both X and Y

2. X\[Cross]Y is also a unit vector

3. If X and Y are unit vectors, but X is not perpendicular to Y, then can X\[Cross]Y be a unit vector? Why not?

Recall that the definition of the cross product is:


\vec a \times \vec b = |a| |b| sin \theta \ \hat n


where \theta is the smaller angle between \vec a and \vec b, and \hat n is a unit vector perpendicular to the plane containing \vec a and \vec b. If the magnitude of \vec a and \vec b are both 1, and \theta is something other than 90 degrees, can the magnitude of the cross product ever be equal to 1?