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In a regualr hexagon ABCDEF, A vecB = ve...

In a regualr hexagon ABCDEF, `A vecB = vec a, B vec C = vecb and C vec D = vec c. " Then " A vec E = `

A

`veca + vec b + vec c`

B

`2 veca + vec b + vec c `

C

`vec a + vec c `

D

`vec a + 2 vec b + 2 vec c `

Text Solution

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To solve the problem, we need to express the vector \( \vec{AE} \) in terms of the vectors \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \) given in the regular hexagon \( ABCDEF \). ### Step-by-Step Solution: 1. **Understanding the Regular Hexagon:** In a regular hexagon, the vertices are equidistant from the center, and the angles between adjacent sides are \( 120^\circ \). We can represent the vertices in a coordinate system for easier vector analysis. 2. **Setting Up the Vectors:** Let's denote the position vectors of the vertices as follows: - \( \vec{A} \) (position vector of point A) - \( \vec{B} \) (position vector of point B) - \( \vec{C} \) (position vector of point C) - \( \vec{D} \) (position vector of point D) - \( \vec{E} \) (position vector of point E) - \( \vec{F} \) (position vector of point F) 3. **Expressing the Given Vectors:** According to the problem: - \( \vec{AB} = \vec{a} \) - \( \vec{BC} = \vec{b} \) - \( \vec{CD} = \vec{c} \) We can express these vectors in terms of position vectors: \[ \vec{B} = \vec{A} + \vec{a} \] \[ \vec{C} = \vec{B} + \vec{b} = \vec{A} + \vec{a} + \vec{b} \] \[ \vec{D} = \vec{C} + \vec{c} = \vec{A} + \vec{a} + \vec{b} + \vec{c} \] 4. **Finding the Position Vector of E:** In a regular hexagon, the vector \( \vec{AE} \) can be expressed as: \[ \vec{E} = \vec{A} + \vec{a} + \vec{b} + \vec{c} + \vec{c} + \vec{b} + \vec{a} \] This is because moving from A to E involves moving through B, C, and D, and then back to E. 5. **Final Expression for \( \vec{AE} \):** The vector \( \vec{AE} \) can be expressed as: \[ \vec{AE} = \vec{E} - \vec{A} = (\vec{A} + \vec{a} + \vec{b} + \vec{c} + \vec{c} + \vec{b} + \vec{a}) - \vec{A} \] Simplifying this gives: \[ \vec{AE} = \vec{a} + \vec{b} + \vec{c} \] ### Conclusion: Thus, the vector \( \vec{AE} \) is given by: \[ \vec{AE} = \vec{a} + \vec{b} + \vec{c} \]
To solve the problem, we need to express the vector \( \vec{AE} \) in terms of the vectors \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \) given in the regular hexagon \( ABCDEF \). ### Step-by-Step Solution: 1. **Understanding the Regular Hexagon:** In a regular hexagon, the vertices are equidistant from the center, and the angles between adjacent sides are \( 120^\circ \). We can represent the vertices in a coordinate system for easier vector analysis. 2. **Setting Up the Vectors:** ...
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