Verifying Orthogonal Curvilinear Coordinate System

[latentcorpse]

For the transformation
$u_1=2x-y$
$u_2=x+2y$
$u_3=3z$

verify that the $u_i$ form an orthogonal curvilinear coordinate system

 

[lanedance]

any ideas? how do you show ui & uj are orthogonal when j does not equal i

 

[latentcorpse]

show dot product is 0.
if $u_1=(2x,-y,0),u_2=(x,2y,0)$ then
$u_1 \cdot u_2 = 2x^2 -2y^2 \neq 0$ though

 

[gabbagabbahey]

Are you sure that the vectors aren't supposed to be:
$$\vec{u_1}=2\hat{x}-\hat{y}$$
$$\vec{u_2}=\hat{x}+2\hat{y}$$
$$\vec{u_2}=3\hat{z}$$

...there is a big difference between the scalar $x$ and the unit vector $$\hat{x}$$!