Coordinates relative to a basis

In summary, the question asks for the x'y-coordinates of points given their xy-coordinates in a rectangular coordinate system, using unit basis vectors i and j and x'y'-coordinate system using unit basis vectors u1 and u2. To find the x'y-coordinates, we can use the equation (v)S = (c1, c2), where c1 and c2 are the solutions to c1u1 + c2u2 = i + j. By finding the values of a1, a2, b1, and b2, any linear combination of i and j can be rewritten in terms of u1 and u2.
  • #1
derryck1234
56
0

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?
 
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  • #2
derryck1234 said:

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

RGV
 
  • #3
Thanks
 

FAQ: Coordinates relative to a basis

What are coordinates relative to a basis?

Coordinates relative to a basis refer to a system of measurement used to describe the position of a point or object in space. It is based on a set of reference points, known as the basis, which serves as the origin for measuring distances and angles.

How are coordinates relative to a basis used in scientific research?

Coordinates relative to a basis are used in various fields of science, such as physics, engineering, and geography, to accurately describe the location and movement of objects or phenomena in space. This allows for precise measurements and calculations to be made.

What is the difference between absolute and relative coordinates?

Absolute coordinates refer to the position of an object in relation to a fixed point, such as the origin of a coordinate system. On the other hand, relative coordinates are measured from a reference point that can change, such as the position of an object in relation to another moving object.

What is the role of a basis in determining coordinates?

The basis serves as the reference point or origin for measuring distances and angles in a coordinate system. It is typically represented by the coordinates (0,0) and is used to determine the position of other points or objects in relation to it.

Can coordinates relative to a basis be used in three-dimensional space?

Yes, coordinates relative to a basis can be used in both two-dimensional and three-dimensional space. In three-dimensional space, an additional axis, usually denoted as the z-axis, is added to the coordinate system to represent the vertical direction.

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