Final answer:
The subset of vectors in the form a 0 c is closed under addition, subtraction, and scalar multiplication, confirming that operations maintain the subset's form. However, it is not closed under the cross product (option d), which can produce a vector not adhering to the form a 0 c.
Step-by-step explanation:
The question whether the subset of vectors in the form a 0 c is closed under a particular operation requires us to understand some vector operation properties. Given this form, where a and c can be any scalars, we are asked to evaluate four types of closure: addition, subtraction, scalar multiplication, and cross product.
Closure under addition implies that the sum of any two vectors within this subset remains within the subset. For example, if we take two vectors of the form a 0 c and a' 0 c', their sum would be (a+a') 0 (c+c'), which is clearly still of the form a 0 c. Therefore, the subset is closed under addition.
Similarly, for subtraction, we can subtract one vector of this form from another to get (a-a') 0 (c-c'), which keeps the result within the subset, indicating closure under subtraction.
When it comes to scalar multiplication, if we take any vector a 0 c and multiply it by some scalar b, we have (ba) 0 (bc). This result is still in the form a 0 c, signifying that the subset is closed under scalar multiplication.
The cross product is a bit tricky; when we cross two vectors of the form a 0 c, the resulting vector will not necessarily have a zero in the middle component, which means the subset is not closed under the cross product.
To directly answer the student's question, we choose only one option that is factually correct based on the evidence presented. Hence, the subset of vectors in the form a 0 c is closed under addition, closed under subtraction, and closed under scalar multiplication, but it is not closed under the cross product. Therefore, options A, B, and C are correct, whereas D is not. Choose only one option, though, would imply we have to mention only one correct option in the final answer. Since the question is multifold, we can state that all options except for D are correct.