Question

ABCD is a parallelogram of vectors, show that 2DC- DB=AC

Answer 1

heya...!!!!


AB = DC = a-----(1)
   
BC = AD = b -----(1)

AC,DB be the diagonals     [a,b are vectors ]

AC+DB

=> (AB+BC)+(DA+DC)

=> (a+b)+(-b+a)  
     [ -b as the direction is negative]
=> 2a

=> 2DC

hence,

2DC- DB=AC

Answer 2

A parallelogram ABCD is drawn in such a way that AB = CD and DA = CB
We have to proof : 2DC - DB = AC

According to triangle rule of summation , we can see that DA + AB = DB ----(1)
and -DA + DC = AC [ because negative vector of DA is -DA ]
⇒DC = AC + DA ------(2)
Now, LHS = 2DC - DB
= 2(AC + DA) - (DA + AB) [ From equation (1) and (2),
= 2AC + 2DA - DA - AB
= 2AC + DA - AB
= 2AC + DA - CD [ ∵AB = CD ]
= 2AC + (-AC) [ from equation (2)]
= 2AC - AC = AC = RHS