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

Options

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

Explanation

To find the position vector of corner D of the parallelogram ABCD given the position vectors of corners A, B, and C, we can follow these steps:

### Step-by-Step Solution:

1. **Define 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
We need to find the position vector of point D, which we will denote as \( \vec{D} \).

2. **Understanding the Properties of a Parallelogram**:
In a parallelogram, the diagonals bisect each other. Therefore, the midpoints of the diagonals AC and BD are the same.

3. **Calculate the Midpoint of Diagonal AC**:
The midpoint of diagonal AC can be calculated as:
\[ \text{Midpoint of AC} = \frac{\vec{A} + \vec{C}}{2} \]

4. **Calculate the Midpoint of Diagonal BD**:
The midpoint of diagonal BD can be calculated as:
\[ \text{Midpoint of BD} = \frac{\vec{B} + \vec{D}}{2} \]

5. **Set the Midpoints Equal**:
Since the midpoints of the diagonals are equal, we can set the two expressions for the midpoints equal to each other:
\[ \frac{\vec{A} + \vec{C}}{2} = \frac{\vec{B} + \vec{D}}{2} \]

6. **Eliminate the Denominator**:
To eliminate the denominator, multiply both sides of the equation by 2:
\[ \vec{A} + \vec{C} = \vec{B} + \vec{D} \]

7. **Rearrange to Solve for \( \vec{D} \)**:
Now, rearranging the equation to isolate \( \vec{D} \):
\[ \vec{D} = \vec{A} + \vec{C} - \vec{B} \]

8. **Final Expression**:
Thus, the position vector of corner D is:
\[ \vec{D} = \vec{A} + \vec{C} - \vec{B} \]