In ur question ,

P₁(x1,y1) and P₂(x2,y2)

OP₁= sqrt(x1²+y1²)

OP₂ =sqrt(x2²+y2²)

slope of OP₁ (m₁)= y1/x1

slope of OP₂ (m₂)= y2/x2

tan (P₁OP₂) = (m₁-m₂)/(1+m₁m₂)

cos(P₁OP₂) = (1+m₁m₂)/(sqrt((m₁-m₂)²+(1+m₁m₂)²))

OP₁.OP₂. cos(P₁OP₂) =sqrt(x1²x2²+x1²y2²+y1²x2²+y1²y2²) . (x1x2+y1y2)/

                                                             sqrt(x1²x2²+x1²y2²+y1²x2²+y1²y2²)

Therefore , OP₁.OP₂.cos (P₁OP₂) = x1x2+y1y2