Final answer:
The set given by the vector operation c(x,y,z) = (0,cy,cz) is not a vector space due to the lack of closure under scalar multiplication, thus the correct answer is C. The essential properties of a vector space, such as associative and commutative properties, are clearly defined and must apply to all vector components.
Step-by-step explanation:
The question pertains to whether a given set satisfies the properties of a vector space. For a set to be a vector space, several conditions must be met, including closure under vector addition and scalar multiplication, the associative and commutative properties of vector addition, and the existence of a multiplicative identity. Given the operation c(x,y,z) = (0,cy,cz), the set does not satisfy the scalar multiplication property of a vector space because multiplying a vector by a scalar should affect all components, not just the latter two. Therefore, the correct choice is C. The set is not closed under scalar multiplication because the scalar c does not affect the x-component of the vector as it should in a vector space.
To verify that the order of addition does not affect the sum of three vectors, one could take vectors A, B, and C and show that A+B+C equals the sum obtained with any other order of addition. This demonstrates the commutative property of vector addition: A+B = B+A. Scalar multiplication by a sum of vectors is distributive, which is a crucial property of vector spaces. While dealing with the components of vectors, we should consider the distributive property to understand operations like the dot product or the cross product.