B Can Shifting Vectors Affect Their Exponential Distance Sum Equality?

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The discussion explores whether the equality of exponential distance sums for two sets of n-dimensional vectors remains valid when a non-zero vector is subtracted from each vector in the sets. It is established that in a one-dimensional case, the equation holds true if the corresponding vectors are equal. The conversation also questions the existence of alternative solutions that satisfy the original equations, particularly when vectors are negated. An example is proposed where the vectors are opposites, prompting further inquiry into the implications of such configurations. The thread ultimately seeks to clarify the conditions under which the exponential distance sum equality is preserved.
thatboi
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Hi all,
Suppose I had some some n-dimensional vectors ##\vec{a}_{1}, \vec{a}_{2}, \vec{b}_{1},\vec{b}_{2}## that satisfied ##e^{||\vec{a}_{1}||^2}+e^{||\vec{a}_{2}||^2}=e^{||\vec{b}_{1}||^2}+e^{||\vec{b}_{2}||^2}##. Now suppose there was another non-zero n-dimensional vector ##\vec{A}##. Is there anything I can say about the equation ##e^{||\vec{a}_{1}-\vec{A}||^2}+e^{||\vec{a}_{2}-\vec{A}||^2}=e^{||\vec{b}_{1}-\vec{A}||^2}+e^{||\vec{b}_{2}-\vec{A}||^2}##? For example, is the equation satisfied for ##\vec{a}_{i} \neq \vec{b}_{j}## for ##i,j = {1,2}##. Also I mean ##||\cdot||## as in the L2 norm.
 
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Can you find an example that fails your equation? Start with a simple (1-dimensional) case.
 
scottdave said:
Can you find an example that fails your equation? Start with a simple (1-dimensional) case.
I suppose even in the 1-dimensional case, my second equation is satisfied as long as ##a_{1}=b_{1}##, ##a_{2}=b_{2}## or vice versa right. I was just wondering if there was some other solutions ##a_{1},b_{1}## that satisfied the set of equations above.
 
thatboi said:
I suppose even in the 1-dimensional case, my second equation is satisfied as long as ##a_{1}=b_{1}##, ##a_{2}=b_{2}## or vice versa right. I was just wondering if there was some other solutions ##a_{1},b_{1}## that satisfied the set of equations above.
What happens when ##a_{1}= -b_{1}##, ##a_{2}= -b_{2}## for example?
 
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