Question

If \( C \) is the midpoint of line segment \( \overline{AB} \), and \( P \) is any point outside the line \( AB \), then:

• A. \( \vec{PA} + \vec{PB} + 2\vec{PC} = \vec{0} \)
• B. \( \vec{PA} + \vec{PB} + \vec{PC} = \vec{0} \)
• C. \( \vec{PA} + \vec{PB} = 2\vec{PC} \)
• D. \( \vec{PA} + \vec{PB} = \vec{PC} \)

Hint:

Midpoint Vectors}
Use vector notation for midpoint: \( C = \frac{A + B}{2} \)
Rearranging helps reveal symmetric vector identities
Helpful in geometric vector proofs and constructions

Answer 1

The Correct Option is C

Let \( C = \frac{A + B}{2} \), and use vector addition: \[ \vec{PC} = \vec{P} - \vec{C} = \vec{P} - \frac{\vec{A} + \vec{B}}{2} \Rightarrow 2\vec{PC} = 2\vec{P} - (\vec{A} + \vec{B}) \] Now: \[ \vec{PA} + \vec{PB} = (\vec{P} - \vec{A}) + (\vec{P} - \vec{B}) = 2\vec{P} - (\vec{A} + \vec{B}) = 2\vec{PC} \]