Question

In a regular hexagon ABCDEF , Show that vector (AB + AC + AD + AE + AF) = 3 vector AD 

Answer 1

Let vector AB = vector a , vector BC = vector b 

vector AC = vector AB + vector BC = vector (a + b)

vector AD = 2 vector BC = 2 vector b 

vector AE = vector (AD + DE) = vector (AD – ED) = 2 vector (b – a)

 vector AF = vector CD = vector (CA + AD) = vector (-AC + AD) = vector -( a + b ) + 2 vector b = vector (-a – b) + 2 vector b = vector (b – a)

Consider, 

vector(AB + AC + AD + AE + AF) = vector (a + ( a + b)) + 2 vector b + vector (2 b – a) + vector (b – a)

= 6 vector b 

= 3 (2 vector b)

= 3 vector AD