What I ask is if $1$ meter in $x$ direction is $2$ times bigger than $1$ meter in $y$ direction. What is the length of hypotenuse when for ex, $3$ in $x$ direction and $4$ in $y$ direction ?
I thought this when i was studying weighted least squares and there uses Mahalanobis distance. It is a very similar idea, but there uses the variance-covariance to compare scales of dimensions. I couldn't directly link variance to exact scale factor like $2$ in this example. I did something but i am not sure if it is right.
++ After thinking, i can rephrase better. Now i think of a moving object that moves with $V$ speed in $y$ direction and $2V$ speed in $x$ direction. If it goes along perpendicular axes, it would take $5.5$ time to move from one corner to another. What is time required if this object moves from one corner to another, diagonally?
Thanks in advance
Answer
One way to interpret what you are asking is to think of a change of coordinates. We have my "normal" coordinate system $(x,y)$ on $\Bbb R^2$ and you have another coordinate system $(z,y)$ with the transformation between your coordinates and mine as $z=2x$. The distance between two points $(z_1,y_1)$ and $(z_2,y_2)$ is $s=\sqrt{4(z_1-z_2)^2+(y_1-y_2)^2}$. It sounds like you are being perverse, but this may be useful. An example would be a crystal where the spacing in one axis is twice the spacing in the other and the coordinates now nicely count lattice positions. You have a space where the metric tensor is $\begin {bmatrix} 4&0\\0&1 \end {bmatrix}$
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