In a quadrilateral ABCD, `vec(AB) + vec(DC) =`

Options

A. `vec(AB) + vec(CB)`
B. `vec(AC) + vec(BD)`
C. `vec(AC) + vec(DB)`
D. `vec(AD) - vec(CB)`

Answer

To solve the problem of finding `vec(AB) + vec(DC)` in the quadrilateral ABCD, we can use the properties of vectors and the relationships between the points in the quadrilateral. Here’s a step-by-step solution:

### Step 1: Define the Position Vectors
Let:
- \( \vec{A} \) be the position vector of point A
- \( \vec{B} \) be the position vector of point B
- \( \vec{C} \) be the position vector of point C
- \( \vec{D} \) be the position vector of point D

### Step 2: Express Vectors AB and DC
The vector \( \vec{AB} \) can be expressed as:
\[ \vec{AB} = \vec{B} - \vec{A} \]
Similarly, the vector \( \vec{DC} \) can be expressed as:
\[ \vec{DC} = \vec{C} - \vec{D} \]

### Step 3: Add the Two Vectors
Now, we need to find \( \vec{AB} + \vec{DC} \):
\[ \vec{AB} + \vec{DC} = (\vec{B} - \vec{A}) + (\vec{C} - \vec{D}) \]

### Step 4: Simplify the Expression
Combine the terms:
\[ \vec{AB} + \vec{DC} = \vec{B} - \vec{A} + \vec{C} - \vec{D} \]
Rearranging gives:
\[ \vec{AB} + \vec{DC} = (\vec{B} + \vec{C}) - (\vec{A} + \vec{D}) \]

### Step 5: Conclusion
Thus, we can conclude that:
\[ \vec{AB} + \vec{DC} = \vec{B} + \vec{C} - \vec{A} - \vec{D} \]

This expression shows the relationship between the vectors in quadrilateral ABCD.