Question

Subtraction of vectors obeys
(A) Commutative law
(B) Associative law
(C) Distributive law
(D) All the above​

Answer 1

Answer:

distributive law

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Answer 2

Answer:

Subtraction of vectors obeys Distributive law.

Explanation:

Subtraction of vectors obeys only Distributive law but It does not obey Communicative law and Associative law.But why??

I will explain it in below.

If a and b are numbers, then subtraction is neither commutative nor associative. Because vector spaces are, in a sense, just number lines pointing in different directions, vector subtraction “inherits” that property.

Subtraction (like division) is a sort of “reverse problem”. This kind of thing happens throughout mathematics. You have some relatively straightforward thing, like addition. You’ve defined how to compute a+b and it’s fairly simple.

For example, with whole (or natural) numbers, you define a+b by saying that a+0=a and a+s(b)=s(a)+b , where s is the “successor” function—the next larger number. So 2+2=3+1=4+0=4 , where the first few steps use the rule a+s(b)=s(a)+b , and the last step uses the rule a+0=a . Addition is “easy”, because it’s based on counting.

Subtraction is more complicated. Instead of giving you a and b and asking for the sum, it gives you the sum and a and asks you to find b . It’s more complicated—there isn’t even always an answer, in whole numbers.

Because it’s “backward” like this, you wouldn’t actually expect that you could add the same number to both 3 and 4 to get 6. If you could, then 3 and 4 aren’t actually different numbers. But that’s what commutativity would require. Subtraction ( a−b) asks what number you’d add to b to get a . It can’t be the same number that you’d add to a to get b unless a and b are the same number—in other words, when a−b=0 .

Associativity fails for essentially the same reason. You can work through the details yourself.

The same thing happens in vector spaces. The difference between two vectors is the step you’d have to take to get from point A to point B. That can’t be in the same direction and distance as the step you’d have to take to get from point B to point A, unless A and B are the same point.