How to split a vector into components in non rectangular cartesian coordinate systems?

Question

In rectangular coordinate system I can split the vectors into components by multiplying the cosine of angle (which the vectors make with the respective axis) with that of the magnitude of the vectors. In the similar way I can get the projection of the vectors on each axis. Similarly I can get the projections of the vectors on non rectangular coordinate system also. But how to split the vectors into it's components in non rectangular coordinate system?

Answer 1

Let's call your vector $v$ and the two "non-rectangular coordinate system" vectors as $w_1, w_2$... ...now:

decompose $v$ over the standard "rectangular coordinate system", for example let's say: $v=3x+1y$

decompose $w_1$ and $w_2$ over the standard "rectangular coordinate system", let's say: $w_1 = 0x+1y$ and $w_2 = 1x+1y$

now: $v=\alpha w_1+ \beta w_2$

wich means:

$3=0\alpha+1\beta$

$1= 1\alpha+1\beta$

solving this system will give you $\alpha$ and $\beta$ so you can write $v=\alpha w_1+\beta w_2$

Answer 2

Suppose you are given a basis of dimension N: $\beta = \{\beta_1,\beta_2, \ldots, \beta_N\}$. We would like to find the coefficients $\{c_1,c_2,\ldots,c_N\}$ such that our vector $v=c_1\beta_1 + c_2\beta_2 + ... + c_N\beta_N$.

Equivalently, we would like to solve the following system: $A c = v$ where $A = \begin{bmatrix} \beta_1 & \beta_2 & \cdots & \beta_N \end{bmatrix}$.

Gaussian elimination does exactly this!