1. Displacement Vector

If I have come variables x,y,d, and i(unit vector) how do I combine this into a displacement vector from an arbitrary origin 0. Or can I just us the general form of a displacement vector and substitute these variables in?

2. also, if i have z = x - xeq where xeq is the equilibrium position does the following differential equation satisfy this condition:

mz'' + rx' +kx = kxeq

this meant to model a forced oscillating system relating to that of an seismograph containing a model spring of force H, damping system of force R and mass of mass M

If I have come variables x,y,d, and i(unit vector) how do I combine this into a displacement vector from an arbitrary origin 0. Or can I just us the general form of a displacement vector and substitute these variables in?
I'm not sure what your variables mean. The displacement vector is defined as
$\Delta \vec{x} = \vec{x} - \vec{x_0}$
where the vectors x and x0 are the final and initial position vectors respectively.

Additionally the position vector is a kind of displacement if you want to think of it that way. The position vector is defined as the vector going from the origin to the position of the object.

-Dan

also, if i have z = x - xeq where xeq is the equilibrium position does the following differential equation satisfy this condition:

mz'' + rx' +kx = kxeq

this meant to model a forced oscillating system relating to that of an seismograph containing a model spring of force H, damping system of force R and mass of mass M
$z = x - x_{eq}$

$z' = x'$

$z'' = x''$

so yes.

-Dan