Subtraction of vectors obeys

Options

A. commutative law
B. Associative law
C. Distributive law
D. All the above

Answer

To solve the question "Subtraction of vectors obeys which law?", we can analyze the properties of vector subtraction in relation to the laws of addition and multiplication. Here’s a step-by-step breakdown:

### Step 1: Understand the Laws of Vectors
Vectors can be added or subtracted, and there are certain mathematical laws that govern these operations. The main laws to consider are:
- Commutative Law
- Associative Law
- Distributive Law

### Step 2: Analyze Commutative Law
The commutative law states that the order of addition does not matter:
\[ A + B = B + A \]

For subtraction, however:
\[ A - B \neq B - A \]
This means that vector subtraction does not obey the commutative law.

### Step 3: Analyze Associative Law
The associative law states that the way in which vectors are grouped does not affect their sum:
\[ (A + B) + C = A + (B + C) \]

For subtraction, we have:
\[ (A - B) - C \neq A - (B - C) \]
This shows that vector subtraction does not obey the associative law either.

### Step 4: Analyze Distributive Law
The distributive law states that multiplication distributes over addition:
\[ C \cdot (A + B) = C \cdot A + C \cdot B \]

For subtraction, we can express it as:
\[ C \cdot (A - B) = C \cdot A - C \cdot B \]
This shows that vector subtraction does obey the distributive law.

### Conclusion
After analyzing all three laws, we conclude that vector subtraction does not obey the commutative or associative laws, but it does obey the distributive law. Therefore, the correct answer is:

**Vector subtraction obeys the distributive law.**

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