Consider the following
If `vec(a) and vec(b)` are the vectors forming consecutive sides of a regular hexagon ABCDEF, then
1. `vec(CE)=vec(b)-2vec(a) " " 2. vec(AE)=2vec(b)-vec(a)`
3. `vec(FA)=vec(a)-vec(b)`
Which of the above are correct?

Options

A. 1 and 2 only
B. 2 and 3 only
C. 1 and 3 only
D. 1, 2, and 3

Answer

D

Explanation

Let ABCDEF be the regular hexagen as shown in the figure.

Let `vec(AB)=vec(a) and vec(BC)=vec(b)`
Join AD, FC and EB. They meet at a common point O,
which is the centre of hexagon.
`AO||BC` so, `vec(AO)=vec(BC)=vec(b)`
`OC||AB ` so, `vec(OC)=vec(AB)=vec(a)`
OAB forms a triangle, `vec(AB)+vec(BO)=vec(AO)`
`rArr vec(BO)=vec(AO)-vec(AB)=vec(b)-vec(a)`
BO=OE and they are on the same line,
so, `vec(BO)=vec(OE)=vec(b)-vec(a)`
In `DeltaOCE, vec(CO) + vec(OE)=vec(CE)`
`rArr vec(CE)=-vec(OC)+OE=-vec(a)+vec(b)-vec(a)=vec(b)-vec(2a)`
So, (1) is correct.
`vec(BE)= 2 vec(OB) ` In `DeltaAEB, vec(AB)+vec(BE)=vec(AE)`
`rArr vec(AE) = vec(AB)+2 vec(BO)+vec(a)+2(vec(b)-vec(a))`
`rArr vec(AE)=vec(a) + 2 vec(b) - 2 vec (a) = 2 vec(b)-vec(a)`
So, (2) is also correct,
`FA||OB rArr vec(FA)=-vec(BO)=-(vec(b)-vec(a))=vec(a)-vec(b)`
So, (3) is also correct.
So, (1) , (2) & (3) are correct.