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Question: f the vectors represented by the sides AB and BC of the regular hexagon ABCDEF be **a** and **b**, t...

f the vectors represented by the sides AB and BC of the regular hexagon ABCDEF be a and b, then the vector represented by AE→\overset{\rightarrow}{AE} will be

A

2b−a2\mathbf{b} - \mathbf{a}

B

b−a\mathbf{b} - \mathbf{a}

C

2a−b2\mathbf{a} - \mathbf{b}

D

a+b\mathbf{a} + \mathbf{b}

Answer

2b−a2\mathbf{b} - \mathbf{a}

Explanation

Solution

As in figure AB→=a,\overset{\rightarrow}{AB} = \mathbf{a}, BC→=b,\overset{\rightarrow}{BC} = \mathbf{b}, so AD→=2b\overset{\rightarrow}{AD} = 2\mathbf{b} and ED→=a\overset{\rightarrow}{ED} = \mathbf{a}.

Now, AE→+ED→=AD→⇒AE→=AD→−ED→=2b−a.\overset{\rightarrow}{AE} + \overset{\rightarrow}{ED} = \overset{\rightarrow}{AD} \Rightarrow \overset{\rightarrow}{AE} = \overset{\rightarrow}{AD} - \overset{\rightarrow}{ED} = 2\mathbf{b} - \mathbf{a}.