# Linearizing vectors using Taylor Series

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SPFF
I am linearizing a vector equation using the first order taylor series expansion. I would like to linearize the equation with respect to both the magnitude of the vector and the direction of the vector.

Does that mean I will have to treat it as a Taylor expansion about two variables, x(direction) and x(magnitude)? Or would linearizing with respect to the vector itself make up for both magnitude and direction?

The magnitude of a vector is not independent of the vector itself. I don't know your precise expression, but note that if you linearize a function of the vector ## (x,y)## with magnitude ##r = \sqrt{x^2 + y^2}## around the vector ##(x_0,y_0)## with magnitude ##r_0 = \sqrt{x_0^2 + y_0^2}## then you will automatically linearize any occurence of ##r## as well, since
\begin{align*} r &\approx r_0 + \frac{\partial r}{\partial x}\Bigr|_{(x,y) = (x_0,y_0)}(x - x_0) + \frac{\partial r}{\partial y}\Bigr|_{(x,y) = (x_0, y_0)}(y - y_0)\\ &= r_0 + \frac{x_0}{r_0}(x - x_0) + \frac{y_0}{r_0}(y - y_0) \end{align*}
neglecting terms of quadratic and higher order. It then also follows, for example, that
$$r^3 \approx r_0^3 + 3 r_0 x_0 (x - x_0) + 3 r_0 y_0 (y - y_0) + \text{h.o.t.}$$

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SPFF

I guess what I'm trying to say is, given some vector r which represents say the position of an object in space, if I wanted to linearize its function with respect to its direction only (i.e. perturb the position with respect to its direction not magnitude), then I could do a Taylor expansion where the derivative is with respect to the unit vector that gives the direction of R but not its magnitude. Now if the function F(r) looks something like GMr/r^3 and I want to linearize with respect to direction and magnitude, would a taylor expansion wrt just the vector r be enough, or would I have to do the taylor expansion with respect to two variable, r and r.
Im leaning towards the latter.

$$\begin{bmatrix} x\\ y\\ z \end{bmatrix} \mapsto (x^2 + y^2 + z^2)^{-\frac{3}{2}} \begin{bmatrix} x\\ y\\ z \end{bmatrix}$$
$$F({\mathbf{r}) = F(\mathbf{r}_0}) +DF(\mathbf{r}_0)\cdot(\mathbf{r} - \mathbf{r}_0) + \text{h.o.t.}$$