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In a plane, each vector has only one orthogonal vector (well, two, if you count the negative of one of them). Are you sure you don't mean the normal vector which is orthogonal but outside the plane (in fact, orthogonal to the plane itself)?
At right angles - in two or more dimensions.
For continuous functions, yes.
Do you mean "equations involving exponential functions"? Yes,
Inverse functions? (not sure what you mean)