# Coordinates relative to a basis

1. Sep 22, 2011

### derryck1234

1. The problem statement, all variables and given/known data

(In textbook, given a figure, I cannot redraw that figure in this applet, so I shall describe the question in words)

I am given a rectangular xy coordinate system determined by the unit basis vectors i and j and an x'y'-coordinate system determined by unit basis vectors u1 and u2. Find the x'y-coordinates of the points whose xy-coordinates are given.

(a) (sqrt3, 1) .. (b) (1, 0) .. (c) (0, 1) .. (d) (a, b)

Okay. u1 is 30 degrees anticlockwise from i, and u2 is directly along j.

2. Relevant equations

(v)S = (c1, c2), where c1 and c2 denote the solutions to c1u1 + c2u2 = i + j.

3. The attempt at a solution

I don't know. I think I'm going to just jump right in and do this (for a):

c1/cos30 = sqrt3
c2 = 1

But this is incorrect?

I don't know what to do? The textbook had no examples like this?

2. Sep 22, 2011

### Ray Vickson

You can write $\mathbf{i} = a_1 \mathbf{u}_1 + a_2 \mathbf{u}_2$ and $\mathbf{j} = b_1 \mathbf{u}_1 + b_2 \mathbf{u}_2 .$ Do you see how to find $a_1, a_2, b_1, b_2$? Now any linear combination of $\mathbf{i} \mbox{ and } \mathbf{j}$ can be immediately re-written in terms of $\mathbf{u}_1 \mbox{ and } \mathbf{u}_2 .$

RGV

3. Sep 28, 2011

Thanks