Question

A, B, C, D are any four points. If E and F are mid points of AC and BD respectively, then \( \vec{AB}+\vec{CB}+\vec{CD}+\vec{AD} = \)

• A. \( \vec{EF} \)
• B. \( 2\vec{EF} \)
• C. \( 3\vec{EF} \)
• D. \( 4\vec{EF} \)

Hint:

Convert geometric vector problems to position vectors to simplify algebra.

Answer 1

The Correct Option is D

Step 1: Vector Approach:

Use position vectors \( \vec{a}, \vec{b}, \vec{c}, \vec{d} \) with respect to an origin. Midpoints: \( \vec{e} = \frac{\vec{a} + \vec{c}}{2} \), \( \vec{f} = \frac{\vec{b} + \vec{d}}{2} \).
Step 2: Detailed Explanation:

Sum \( S = \vec{AB} + \vec{CB} + \vec{CD} + \vec{AD} \). \[ S = (\vec{b} - \vec{a}) + (\vec{b} - \vec{c}) + (\vec{d} - \vec{c}) + (\vec{d} - \vec{a}) \] \[ S = 2\vec{b} + 2\vec{d} - 2\vec{a} - 2\vec{c} \] \[ S = 2(\vec{b} + \vec{d}) - 2(\vec{a} + \vec{c}) \] Substitute \( \vec{b} + \vec{d} = 2\vec{f} \) and \( \vec{a} + \vec{c} = 2\vec{e} \): \[ S = 2(2\vec{f}) - 2(2\vec{e}) = 4(\vec{f} - \vec{e}) = 4\vec{EF} \]
Step 3: Final Answer:

The sum is \( 4\vec{EF} \).