ABCD is a quadrilateral whose diagonals are AC and BD. Which one of the following is correct?

Options

A. `vec(BA)+vec(CD)=vec(AC)+vec(DB)`
B. `vec(BA)+vec(CD)=vec(BD)+vec(CA)`
C. `vec(BA)+vec(CD)=vec(AC)+vec(BD)`
D. `vec(BA)+vec(CD)=vec(BC)+vec(AD)`

Explanation

To solve the problem, we need to analyze the relationships between the sides and diagonals of the quadrilateral ABCD using vector addition. Here’s a step-by-step solution:

### Step 1: Understand the Quadrilateral and its Diagonals
We have a quadrilateral ABCD with diagonals AC and BD. We need to explore the relationships between the sides and diagonals using vector addition.

**Hint:** Visualize the quadrilateral and label the points clearly.

### Step 2: Identify the Sides and Diagonals
The sides of the quadrilateral are AB, BC, CD, and DA. The diagonals are AC and BD. We will use the triangle law of vector addition to find relationships.

**Hint:** Recall that the triangle law states that if two sides of a triangle are represented as vectors, the third side can be represented as the sum of the first two vectors.

### Step 3: Apply Triangle Law of Vector Addition
Consider triangle ACD:
- According to the triangle law, we can write:
\[ \vec{AC} = \vec{AD} + \vec{DC} \]
Rearranging gives:
\[ \vec{DC} + \vec{AD} = \vec{AC} \]

**Hint:** Think about how you can rearrange the vectors to express relationships.

### Step 4: Consider Triangle ABD
Now, consider triangle ABD:
- Again, using the triangle law, we can write:
\[ \vec{AB} + \vec{AD} = \vec{BD} \]
Rearranging gives:
\[ \vec{AB} + \vec{AD} = \vec{BD} \]

**Hint:** Use the same approach as before to express the relationship in terms of the sides and diagonals.

### Step 5: Combine the Results
Now, we can combine the results from both triangles:
1. From triangle ACD:
\[ \vec{DC} + \vec{AD} = \vec{AC} \]
2. From triangle ABD:
\[ \vec{AB} + \vec{AD} = \vec{BD} \]

Adding these two equations gives:
\[ \vec{DC} + \vec{AB} + 2\vec{AD} = \vec{AC} + \vec{BD} \]

**Hint:** Look for common terms in both equations that can help simplify the expression.

### Step 6: Analyze the Options
Now we need to analyze the options provided:
1. \( \vec{BA} + \vec{CD} = \vec{AC} + \vec{BD} \)
2. \( \vec{BA} + \vec{CD} = \vec{BD} \)
3. \( \vec{BA} + \vec{CD} = \vec{AC} + \vec{BD} \)
4. \( \vec{BA} + \vec{CD} = \vec{BC} + \vec{AD} \)

From our derived equations, we can see that the correct relationship is:
\[ \vec{BA} + \vec{CD} = \vec{AC} + \vec{BD} \]

**Hint:** Compare your derived relationships with the options carefully.

### Final Answer
The correct option is:
**B) \( \vec{BA} + \vec{CD} = \vec{AC} + \vec{BD} \)**