<p>If A, B, C, D and E are five coplanar points, then \(\vec{DA} + \vec{DB} + \vec{DC} + \vec{AE} + \vec{BE} + \vec{CE}\) is equal to</p>
Step-by-Step Solution
Key Concept: Regroup the given vector sum by pairing terms strategically: group vectors from D and vectors from E, then express each group in terms of position vectors relative to a common point. The key is recognizing that the sum can be rewritten as 3(DA + DB + DC) - 3(EA + EB + EC), which simplifies using the relationship between these sums.
Step 1: Rewrite the given expression We need to find: $\vec{DA} + \vec{DB} + \vec{DC} + \vec{AE} + \vec{BE} + \vec{CE}$ Rewrite using vector relations: $\vec{AE} = \vec{AD} + \vec{DE}$, $\vec{BE} = \vec{BD} + \vec{DE}$, $\vec{CE} = \vec{CD} + \vec{DE}$ Step 2: Substitute the rewritten forms $\vec{DA} + \vec{DB} + \vec{DC} + (\vec{AD} + \vec{DE}) + (\vec{BD} + \vec{DE}) + (\vec{CD} + \vec{DE})$ Step 3: Recognize that $\vec{AD} = -\vec{DA}$, $\vec{BD} = -\vec{DB}$, $\vec{CD} = -\vec{DC}$ $= \vec{DA} + \vec{DB} + \vec{DC} - \vec{DA} - \vec{DB} - \vec{DC} + 3\vec{DE}$ Step 4: Simplify by canceling terms The terms $(\vec{DA} - \vec{DA}) + (\vec{DB} - \vec{DB}) + (\vec{DC} - \vec{DC}) = \vec{0}$ Therefore: $\vec{0} + 3\vec{DE} = 3\vec{DE}$ ∴ Answer: B
Correct Answer: B