If `vec(a), vec(b), vec(c ), vec(d)` respectively, are position vectors representing the vertices A, B, C, D of a parallelogram, then write `vec(d)` in terms of `vec(a), vec(b)` and `vec(c )`.

Answer

To find the position vector \(\vec{d}\) in terms of the position vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) for the vertices \(A\), \(B\), \(C\), and \(D\) of a parallelogram, we can follow these steps:

### Step 1: Understand the properties of a parallelogram
In a parallelogram, the diagonals bisect each other. This means that the midpoint of diagonal \(AC\) is the same as the midpoint of diagonal \(BD\).

### Step 2: Write the midpoint of diagonal \(AC\)
The midpoint \(O\) of diagonal \(AC\) can be expressed using the position vectors of points \(A\) and \(C\):
\[ \vec{O} = \frac{\vec{a} + \vec{c}}{2} \]

### Step 3: Write the midpoint of diagonal \(BD\)
Similarly, the midpoint \(O\) of diagonal \(BD\) can be expressed using the position vectors of points \(B\) and \(D\):
\[ \vec{O} = \frac{\vec{b} + \vec{d}}{2} \]

### Step 4: Set the two expressions for \(O\) equal to each other
Since both expressions represent the same point \(O\), we can set them equal:
\[ \frac{\vec{a} + \vec{c}}{2} = \frac{\vec{b} + \vec{d}}{2} \]

### Step 5: Eliminate the fraction
To eliminate the fraction, multiply both sides by 2:
\[ \vec{a} + \vec{c} = \vec{b} + \vec{d} \]

### Step 6: Solve for \(\vec{d}\)
Now, we can rearrange the equation to express \(\vec{d}\) in terms of \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\):
\[ \vec{d} = \vec{a} + \vec{c} - \vec{b} \]

### Final Result
Thus, the position vector \(\vec{d}\) can be expressed as:
\[ \vec{d} = \vec{a} + \vec{c} - \vec{b} \]

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