If M and N are the mid-points of the diagonals AC and BD respectively of a quadrilateral ABCD, then the of `vec (AB)+ vec(AD)+vec(CB)+vec(CD)` equals

Options

A. 2 MN
B. 2NM
C. 4 MN
D. 4 NM

Answer

C

Explanation

Let the position vectors of A,B,C,D, M and N are
a,b,c,d, m and n.
Since, M and N are the mid-points of AC and BD.
`therefore m=(a+c)/(2),n=(b+d)/(2)`
Now, `AB+AD+CD+CD`
`=(b-a)+(d-a)+(b-c)+(d-c)`
`=2(b+d)-2)a+c)`
`=2xx2n-2xx2m`
`=4 (n-m)=4 NM`