If `vec(a), vec(b), vec(c )` are the position vectors of corners A, B, C of a parallelogram ABCD. Then what is the position vector of the corner D?

Options

A. A) `vec(a) +vec(b)+ vec(c )`
B. B) `vec(a) +vec(b) - vec(c )`
C. C) `vec(a)- vec(b) +vec(c )`
D. D) `-vec(a) +vec(b) +vec(c )`

Answer

To find the position vector of corner D in the parallelogram ABCD, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Position Vectors**:
Let the position vectors of points A, B, and C be represented as:
- \( \vec{A} \) for point A
- \( \vec{B} \) for point B
- \( \vec{C} \) for point C

2. **Understand the Properties of a Parallelogram**:
In a parallelogram, opposite sides are equal and parallel. Therefore, the vector from A to B (\( \vec{AB} \)) is equal to the vector from C to D (\( \vec{CD} \)):
\[ \vec{AB} = \vec{CD} \]

3. **Express the Vectors**:
The vector \( \vec{AB} \) can be expressed as:
\[ \vec{AB} = \vec{B} - \vec{A} \]
Similarly, the vector \( \vec{CD} \) can be expressed as:
\[ \vec{CD} = \vec{D} - \vec{C} \]

4. **Set the Two Expressions Equal**:
Since \( \vec{AB} = \vec{CD} \), we can set the two expressions equal to each other:
\[ \vec{B} - \vec{A} = \vec{D} - \vec{C} \]

5. **Rearrange to Solve for \( \vec{D} \)**:
Rearranging the equation gives:
\[ \vec{D} = \vec{B} - \vec{A} + \vec{C} \]
This can also be rearranged to:
\[ \vec{D} = \vec{A} + \vec{C} - \vec{B} \]

6. **Final Expression for \( \vec{D} \)**:
Thus, the position vector of corner D is given by:
\[ \vec{D} = \vec{A} + \vec{C} - \vec{B} \]

### Final Answer:
The position vector of corner D is:
\[ \vec{D} = \vec{A} + \vec{C} - \vec{B} \]