Coordinates relative to a basis

[derryck1234]

Homework Statement

(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.

Homework Equations

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

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?

 

[Ray Vickson]

Homework Statement

(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.

Homework Equations

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

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?

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

 

[derryck1234]

Thanks