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Greek Civilisation

Five forces A...

Question

Five forces $\vec{A B,} \vec{A C,} \vec{A D,} \vec{A E}$ and $\vec{A F}$ act at the vertex of a regular hexagon ABCDEF. Prove that the resultant is
6 $\vec{A O,}$ where O is the centre of hexagon.

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Solution

$\vec{A B} + \vec{A C} + \vec{A D} + \vec{A E} + \vec{A F}$

Consider âˆ†ADE,

$\begin{array}{l} \vec{A D} + \vec{D E} + \vec{E A} = 0 \\ \vec{A D} + \vec{D E} = \vec{A E} \\ 2 \vec{A O} + (- \vec{A B}) = \vec{A E} [\vec{A D} = 2 \vec{A O} and E D | | A B \vec{D E} = - \vec{A B}] \\ ∴ \vec{A E} + \vec{A B} = 2 \vec{A O} . . . . . (1) \end{array}$

Now, consider âˆ†ADC

$\vec{A C} + \vec{C D} + \vec{D A} = 0 \vec{A C} + \vec{C D} = \vec{A D} [∵ \vec{C D} = \vec{A F}] \vec{A C} + \vec{A F} = 2 \vec{A O} . . . . . (2)$

Using (1) and (2),

$\vec{A B} + \vec{A E} + \vec{A C} + \vec{A F} + \vec{A D} 2 \vec{A O} + 2 \vec{A O} + 2 \vec{A O} = 6 \vec{A O}$

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Similar questions

\vec{A B,} \vec{A C,} \vec{A D,} \vec{A E}

\vec{A F}

act at the vertex of a regular hexagon ABCDEF. Prove that the resultant is
6

\vec{A O,}

where O is the centre of hexagon.

Q. Consider the regular hexagon

A B C D E F

O

\to A B, \to A C, \to A D, \to A E, \to A F

act at the vertex

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A B C D E F

, then their resultant is

Q. If Five forces

\to A B, \to A C, \to A D, \to A E, \to A F

act at the vertex A of a regular hexagon ABCDEF, then their resultant is? (where O is the centroid of the hexagon)

Q. In a regular hexagon

A B C D E F

O

(origin). Prove
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\to A B, \to A C, \to A D, \to A E, \to A F

act at the vertex

A

of regular hexagon

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- - \to A B, - - \to A C, - - \to A D, - - \to A E, - - \to A F

act at the vertex

A

of a regular hexagon

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O

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