If a, b, c be non-zero vectors, then which of the following statements are correct

Options

A. `axx(b-c) = (c-b) xxa `
B. `a.(b + c) = -(b + c).a`
C. `a xx(b + c) = (c + b) xx a`
D. `a.(b-c) = (c-b) .a`

Answer

To determine which statements regarding the non-zero vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are correct, we will analyze each option step by step.

### Step 1: Analyze Option 1 - \( \mathbf{a} \times \mathbf{b} - \mathbf{c} \)

We need to check if \( \mathbf{a} \times \mathbf{b} - \mathbf{c} \) holds true.

Using the property of the cross product:
\[ \mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a}) \]
This implies that the cross product is anti-commutative.

Thus, we can express:
\[ \mathbf{a} \times \mathbf{b} - \mathbf{c} = -(\mathbf{b} \times \mathbf{a}) - \mathbf{c} \]

This statement can be rearranged, but it does not provide any contradiction, so we can conclude that this option is correct.

### Step 2: Analyze Option 2 - \( \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) \)

We need to check if \( \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) \) is equal to \( \mathbf{b} + \mathbf{c} \cdot \mathbf{a} \).

Using the distributive property of the dot product:
\[ \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c} \]

This is equal to \( \mathbf{b} \cdot \mathbf{a} + \mathbf{c} \cdot \mathbf{a} \) (since dot product is commutative).

Thus, this option is also correct.

### Step 3: Analyze Option 3 - \( \mathbf{a} \times (\mathbf{b} + \mathbf{c}) \)

We need to check if \( \mathbf{a} \times (\mathbf{b} + \mathbf{c}) \) is equal to \( \mathbf{b} + \mathbf{c} \times \mathbf{a} \).

Using the distributive property of the cross product:
\[ \mathbf{a} \times (\mathbf{b} + \mathbf{c}) = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c} \]

However, \( \mathbf{b} + \mathbf{c} \times \mathbf{a} \) does not equal \( \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c} \) as the cross product is not commutative.

Thus, this option is incorrect.

### Step 4: Analyze Option 4 - \( \mathbf{a} \cdot (\mathbf{b} - \mathbf{c}) \)

We need to check if \( \mathbf{a} \cdot (\mathbf{b} - \mathbf{c}) \) is equal to \( \mathbf{b} - \mathbf{c} \cdot \mathbf{a} \).

Using the distributive property of the dot product:
\[ \mathbf{a} \cdot (\mathbf{b} - \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} - \mathbf{a} \cdot \mathbf{c} \]

This is not equal to \( \mathbf{b} \cdot \mathbf{a} - \mathbf{c} \cdot \mathbf{a} \) because the signs do not match.

Thus, this option is incorrect.

### Conclusion

From the analysis above, the correct options are:
- Option 1: Correct
- Option 2: Correct
- Option 3: Incorrect
- Option 4: Incorrect