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Question

If ABCDEF is a regular hexagon and if ¯¯¯¯¯¯¯¯AB+¯¯¯¯¯¯¯¯AC+¯¯¯¯¯¯¯¯¯AD+¯¯¯¯¯¯¯¯AE+¯¯¯¯¯¯¯¯AF=λ¯¯¯¯¯¯¯¯¯AD then λ is

A
0
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B
1
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C
2
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D
3
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Solution

The correct option is A 3
ABCDEF is a regular hexagon.
¯¯¯¯¯¯¯¯AB+¯¯¯¯¯¯¯¯AC+¯¯¯¯¯¯¯¯¯AD+¯¯¯¯¯¯¯¯AE+¯¯¯¯¯¯¯¯AF
=¯¯¯¯¯¯¯¯AB+¯¯¯¯¯¯¯¯AB+¯¯¯¯¯¯¯¯BC+¯¯¯¯¯¯¯¯¯AD+¯¯¯¯¯¯¯¯AE+¯¯¯¯¯¯¯¯AE+¯¯¯¯¯¯¯¯EF
=¯¯¯¯¯¯¯¯AB+¯¯¯¯¯¯¯¯AB+¯¯¯¯¯¯¯¯¯AD+¯¯¯¯¯¯¯¯AE+¯¯¯¯¯¯¯¯AE (¯¯¯¯¯¯¯¯BC+¯¯¯¯¯¯¯¯EF=0)
=3¯¯¯¯¯¯¯¯¯AD+2(¯¯¯¯¯¯¯¯AB+¯¯¯¯¯¯¯¯¯DE)
=3¯¯¯¯¯¯¯¯¯AD (¯¯¯¯¯¯¯¯AB+¯¯¯¯¯¯¯¯¯DE=0)
λ=3
Hence, option D.

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