let a,b,c be nth dimensional vector
a= (a1,a2,...,an)
b= (b1,b2,...,bn)
c= (c1,c2,...,cn)
a.(b+c)
=(a1,a2,...,an).(b1+c1,b2+c2,...,bn+cn)
=a1(b1+c1)+a2(b2+c2)+....+an(bn+cn)
=(a1b1+a2b2+..+anbn) + (a1c1+a2c2+..+ancn)
=a.b+b.c
now
a(x1,y1),b(x2,y2),
a.b
=(x1,y1).(x2,y2)
= x1x2+y1y2
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