Final answer:
To form an orthonormal set, vectors must be orthogonal and normalized to unit magnitude. Normalization does not affect orthogonality, and vector addition is not the method to create an orthonormal set. Option C is correct.
Step-by-step explanation:
The task of normalizing a set to produce an orthonormal set involves two steps: ensuring that each vector is orthogonal to the others, and normalizing each vector such that it has a magnitude of one. Vectors are orthogonal if they are perpendicular to each other, which means their direction angles differ by 90°, resulting in a scalar product of zero.
The process of normalization scales a vector to unit length by dividing it by its magnitude, thereby not affecting its direction and hence not impacting its orthogonality with other vectors. The claim that normalization doesn't impact the orthogonality of a set is correct (C). Normalization alone doesn't achieve orthonormalization, as it must be paired with ensuring the vectors are orthogonal; hence, option D is incorrect.
Orthonormal sets have the additional property that the vectors are not only orthogonal but also have unit magnitude after normalization (A and B). Together, these properties allow for simple calculations in many mathematical applications.
It's also important to note that while vectors can be multiplied by scalars, and vectors can be added together, the operation to achieve an orthonormal set does not involve vector addition as stated in option D. Instead, orthonormalization might involve a process such as the Gram-Schmidt procedure, which systematically orthonormalizes a set of vectors.