Understanding the Problem: Vectors in a Parallelogram

The question asks for the vector sum of the position vectors of the vertices of a parallelogram ABCD from a given origin O. We are given that P is the point where the diagonals of the parallelogram intersect.

Key Properties of Parallelograms and Vectors

A fundamental property of a parallelogram is that its diagonals bisect each other. This means the point of intersection of the diagonals, P, is the midpoint of both diagonal AC and diagonal BD.

In vector form, if P is the midpoint of a line segment AB, and O is the origin, the position vector of P, \(\overrightarrow {{\rm{OP}}} \), is given by the midpoint formula:

\(\overrightarrow {{\rm{OP}}} = \frac{\overrightarrow {{\rm{OA}}} + \overrightarrow {{\rm{OB}}} }{2}\)

where \(\overrightarrow {{\rm{OA}}} \) and \(\overrightarrow {{\rm{OB}}} \) are the position vectors of points A and B respectively, relative to the origin O.

Applying Vector Properties to the Parallelogram

Since P is the midpoint of diagonal AC, we can write the vector relationship:

\(\overrightarrow {{\rm{OP}}} = \frac{\overrightarrow {{\rm{OA}}} + \overrightarrow {{\rm{OC}}} }{2}\)

Multiplying both sides by 2, we get:

Equation 1: \(\overrightarrow {{\rm{OA}}} + \overrightarrow {{\rm{OC}}} = 2\overrightarrow {{\rm{OP}}} \)

Similarly, since P is also the midpoint of diagonal BD, we can write the vector relationship:

\(\overrightarrow {{\rm{OP}}} = \frac{\overrightarrow {{\rm{OB}}} + \overrightarrow {{\rm{OD}}} }{2}\)

Multiplying both sides by 2, we get:

Equation 2: \(\overrightarrow {{\rm{OB}}} + \overrightarrow {{\rm{OD}}} = 2\overrightarrow {{\rm{OP}}} \)

Calculating the Required Vector Sum

We need to find the value of \(\overrightarrow {{\rm{OA}}} + \overrightarrow {{\rm{OB}}} + \overrightarrow {{\rm{OC}}} + \overrightarrow {{\rm{OD}}} \).

We can group the terms in the sum:

\((\overrightarrow {{\rm{OA}}} + \overrightarrow {{\rm{OC}}} ) + (\overrightarrow {{\rm{OB}}} + \overrightarrow {{\rm{OD}}} )\)

Now, we can substitute the results from Equation 1 and Equation 2 into this expression:

\((2\overrightarrow {{\rm{OP}}} ) + (2\overrightarrow {{\rm{OP}}} )\)

Adding these two terms gives:

\(4\overrightarrow {{\rm{OP}}} \)

Therefore, the vector sum \(\overrightarrow {{\rm{OA}}} + \overrightarrow {{\rm{OB}}} + \overrightarrow {{\rm{OC}}} + \overrightarrow {{\rm{OD}}} \) is equal to \(4\overrightarrow {{\rm{OP}}} \).

This confirms that the sum of the position vectors of the vertices of the parallelogram from an origin O is four times the position vector of the intersection of its diagonals from the same origin O.

Revision Table: Vectors in Parallelograms

Let's quickly recap the key concepts used:

• Parallelogram Properties: Diagonals bisect each other at their midpoint.
• Position Vector: A vector representing the position of a point relative to an origin.
• Midpoint Formula (Vectors): The position vector of the midpoint of a segment is the average of the position vectors of its endpoints. If M is the midpoint of AB and O is the origin, \(\overrightarrow{OM} = \frac{\overrightarrow{OA} + \overrightarrow{OB}}{2}\).

Additional Information: Vector Applications

Vector geometry is a powerful tool for solving geometric problems. Representing points and geometric figures using vectors simplifies calculations, especially involving position, displacement, and relative locations. The property used here for parallelograms is a classic example of how vector addition and the midpoint formula can be applied to prove geometric properties or find relationships between points.

For any parallelogram ABCD, we know that \(\overrightarrow{AB} = \overrightarrow{DC}\). If we choose one vertex, say A, as the origin, then the position vectors are \(\overrightarrow{AA} = \overrightarrow{0}\), \(\overrightarrow{AB} = \mathbf{b}\), \(\overrightarrow{AD} = \mathbf{d}\), and \(\overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AB} + \overrightarrow{AD} = \mathbf{b} + \mathbf{d}\) (since \(\overrightarrow{BC} = \overrightarrow{AD}\) in a parallelogram). The diagonals are AC and BD. The midpoint of AC is \(\frac{\overrightarrow{AA} + \overrightarrow{AC}}{2} = \frac{\overrightarrow{0} + \mathbf{b} + \mathbf{d}}{2} = \frac{\mathbf{b} + \mathbf{d}}{2}\). The midpoint of BD is \(\frac{\overrightarrow{AB} + \overrightarrow{AD}}{2} = \frac{\mathbf{b} + \mathbf{d}}{2}\). This shows that the midpoints coincide, proving the diagonals bisect each other using vectors, even without an external origin O.