Vector component in different co ordinate system

[darwined]

I am facing difficulty understanding the vecotr component in the 2 different co ordinate systems. Kindly help me.

Suppose I have a vector A on the 2d co ordinate system x and y and therefore vector A = Axi + Ayj where i and j are unit vectors along x and y axis.

Suppose a new co ordinate system is introduced in which is x'y' and is titlted from the xy plane by angle theta and both the co ordinate systems (xy and x'y') share the same origin.

The component of the vector seem to change on x'y' plane.

A = i'Ax' + j'Ay' (i' and j' are the unit vectors along the x' and y')

and now i' = i cos (theta) + j sin (theta)
j' = - i sin (theta) + j cos (theta)

Kindly help me understand the concept.

 

[Simon Bridge]

What don't you understand about it?

Note:
The vector itself is just an arrow - it does not depend on the axes you draw.
You can put the origin of the axis anywhere, and you can orient the axis however you like.

Try this: first draw an arrow on a bit of paper.
then draw a set of x-y axes also on that paper.
Work out the representation of the vector according to those axes.
How did you do that?

Now draw another set of axes that are in a different orientation.
Work out the representation of the vector according to the new axes - do not use the formula, just use the same method you did before.
See?

 

[darwined]

Thank you for your response.

I have attached the pic that I have worked out.

I am not able to understand how to get the trignonometric expression in place.

 

[Simon Bridge]

Did you try just doing what I said?

 

[darwined]

I guess I tried doing what you suggested. Please correct me if I am wrong.

 

[Simon Bridge]

For the first coordinate system - how did you determine the vector representation?

 

[darwined]

I just read somewhere in some book and I remember it, also it quite intuitive, componets of a vector on the x y plane is simply the measurement of the contribution of the x and y-axis multiplied by their unit vectors respectively.

 

[Simon Bridge]

Great - so do that for the vector you drew on the first set of axes you drew.

 

[darwined]

The first set of axes is made up of xy plane and the unit vectors along x and y direction are i and j. Therefore

A=Axi+Ayj

Now I am confused how to do it for the x'y' plane. Also how to develop trigonometric expression from it.

 

[Simon Bridge]

You are still not following the suggestion.
Maybe I am not being clear. I am trying to get you to focus on the act of representing a vector in an arbitrary coordinate system: what you do when you write down the numerical form of the vector.

Draw an arrow (use a ruler).

Make just one coordinate axes set.
Mark the axes out in 1cm intervals.
Number the intervals 0,1,2,3... from the origin.
Use those numbers to get a representation of your arrow (vector) in that coordinate system.
The answer you get should be a pair of numbers.

If we call this coordinate system A, then you can write $[\vec v]_A = x_A\hat \imath_A + y_A\hat\jmath_A$

Note: it may help to draw the coordinate axes on a transparency or tracing paper.

However - I am only guessing about what it is you don't understand - you have not actually told me yet.
I could be going about this all wrong.
Maybe all you need to realize is that rotating the coordinate axes by some angle is the same as rotating the vector the same angle but in the opposite direction?