Vector a = (x1, Y1), vector b = (X2, Y2), a + B = (x1 + X2, Y1 + Y2), how to express a * B, is it equal to a + B

a*b=(x1,y1)*(x2,y2)
=(x1i+y1j)*(x2i+y2j)
=x1*x2*i^2+x1*y2*(i*j)+y1*x2(i*j)+y1*y2*j^2 (I*j=J*i=0,i^2=1,j^2=1)
=x1*x2+y1*y2
So a * b = X1 * x2 + Y1 * Y2

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