The relation of correlation coefficient of the sum of two vectors.

Question

Does the correlation coefficient of the sum of two vectors between the correlation coefficient of each of them. Suppose I have three vectors $x_1,x_2,x_3$. The correlation coefficient of $x_1$ and $x_2$ is $c_1$, the correlation coefficient of $x_1$ and $x_3$ is $c_2$, the correlation coefficient of $x_1$ and $x_2+x_3$ is $c_3$. Can I state that $$\min(c_1,c_2)<c_3<\max(c_1,c_2)$$

Answer 1

For a short answer, no, you couln't conclude that. For a simple example, let: $$ x_{1}=(1,2,3,4)^{T} $$ $$ x_{2}=(0,3,2,5)^{T} $$ $$ x_{3}=(2,1,4,3)^{T} $$ It's easy to compute: $c_{1}=0.8682$, $c_{2}=0.6$, $c_{3}=1$.