If ABCD is a square, then
`bar(AB) + 2 bar(BC) + 3 bar(CD) + 4 bar(DA)` =

Options

A. 4 `bar(AC)`
B. 3 `bar(AC)`
C. 2 `bar(CA)`
D. `bar(AC)`

Answer

To solve the problem, we need to evaluate the expression \( \bar{AB} + 2 \bar{BC} + 3 \bar{CD} + 4 \bar{DA} \) for a square ABCD.

### Step-by-Step Solution:

1. **Identify the Vectors**:
- Let the length of each side of the square be \( a \).
- The vectors can be represented as:
- \( \bar{AB} = a \hat{i} \) (assuming A is at the origin and B is to the right)
- \( \bar{BC} = a \hat{j} \) (B to C is upwards)
- \( \bar{CD} = -a \hat{i} \) (C to D is to the left)
- \( \bar{DA} = -a \hat{j} \) (D to A is downwards)

2. **Substituting the Vectors into the Expression**:
- Substitute the vectors into the expression:
\[ \bar{AB} + 2 \bar{BC} + 3 \bar{CD} + 4 \bar{DA} = a \hat{i} + 2(a \hat{j}) + 3(-a \hat{i}) + 4(-a \hat{j}) \]

3. **Simplifying the Expression**:
- Combine the terms:
\[ = a \hat{i} + 2a \hat{j} - 3a \hat{i} - 4a \hat{j} \]
- Group the \( \hat{i} \) and \( \hat{j} \) components:
\[ = (a - 3a) \hat{i} + (2a - 4a) \hat{j} \]
\[ = -2a \hat{i} - 2a \hat{j} \]

4. **Factoring Out Common Terms**:
- Factor out \(-2a\):
\[ = -2a (\hat{i} + \hat{j}) \]

5. **Final Result**:
- The final result can be expressed as:
\[ = -2a (\hat{i} + \hat{j}) \]

### Conclusion:
The expression \( \bar{AB} + 2 \bar{BC} + 3 \bar{CD} + 4 \bar{DA} \) simplifies to \( -2a (\hat{i} + \hat{j}) \).