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Addition of Vectors

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Question

If $\to a$ and $\to b$ are vectors represented by two adjacent sides of a regular hexagon, then find vectors represented by other sides.

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Solution

Let $\to A B = \to a$
$\to B C = \to b$
$\Rightarrow \to D E = - \to A B = - \to a$
$\to E F = - \to B C = \to - b$
$\to A B + \to B C = \to A C \Rightarrow \to A C = \to a + \to b$
$\to A C + \to C D = \to D E$
$\to C D = (\to D E) - (\to A C)$
$= (\to - a) - (\to a + b)$
$\to C D = - 2 \to a - \to b$
$\to F A = - \to C D = 2 \to a + \to b$
$∴$ the sides are $(\to a, \to b, - 2 \to a - \to b, - \to a, - \to b, 2 \to a + \to b)$

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