Coordinate System's influence on $L$ distances (Manhattan and Euclidean)
For example, consider the green line. What is its length? In $L_2$, the answer is $1$, in $L_1$, the answer is $1$ as well. 

Now, for the same line, let's rotate it $45^\circ$ counterclockwise. What is the length again? In $L_2$, its length remains to be $1$. However, in $L_1$, using Manhattan distance, it's length is now $\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}=\sqrt2$. As we can see, rotation can change $L_1$ distance.

Rather than rotating the line, we could also equivalently rotated our $x$ and $y$-axis instead.