Vector Algebra
Vector Addition
Grade 12
Question:
<p>P is the point of intersection of the diagonals of the parallelogram ABCD. If O is any point, then $\vec{OA} + \vec{OB} + \vec{OC} + \vec{OD}$ is equal to</p>
<p>(a) $\vec{OP}$</p>
<p>(b) $2\vec{OP}$</p>
<p>(c) $3\vec{OP}$</p>
<p>(d) $4\vec{OP}$</p>
Step-by-Step Solution
Key Concept: The diagonals of a parallelogram bisect each other. Use the midpoint property to express the sum of position vectors.
Solution: We know that P is the mid-point of both AC and BD in a parallelogram. Therefore: $\vec{OA} + \vec{OC} = 2\vec{OP}$ ... (i) And: $\vec{OB} + \vec{OD} = 2\vec{OP}$ ... (ii) Adding (i) and (ii): $\vec{OA} + \vec{OB} + \vec{OC} + \vec{OD} = 4\vec{OP}$
Correct Answer: d