Question

Give the formulae for the transformation of the coordinates of a vector. Plssssssss

Answer 1

sorry, I have no idea

Answer 2

Ha k. Cn any1 else help me.?

Answer 3

transformations into what?

Answer 4

are you talking about linear transformations: R2 -> R3 ??

Answer 5

does this help? http://en.wikipedia.org/wiki/Transformation_matrix

Answer 6

It's a part of a question. Let me type the whole question.

Answer 7

Pls see the attachment for the full question.

Answer 8

Part of forming a basis is that the vectors u,v,w have to be linearly independent. That means that each of the vectors can't be a linear combination of the others. So, let's look at the new vectors (let's call them x,y,z)

x = u+v
y = -u+w
z = u+v+w

If you take any of x,y and z, and linearly combine them, can you end up with another one of x, y or z (hint: you can't)? If not, x, y and z form a basis.

As for the second part of the question, are we talking affine transformation? Those are most easily expressed in a matrix, which reduces the transform to a vector-matrix multiply (which itself is just a series of dot products).
I'm not sure what formulae are being looked for here, because it depends on the transform asked for. If it's just a rotation, it's relatively simple - you can just build a matrix like so:

\[\left[\begin{matrix}\cos \beta \cos \gamma & -\cos \alpha \sin \gamma + \sin \alpha \sin \beta \cos \gamma & \sin \alpha \sin \gamma + \cos \alpha \sin \beta \cos \gamma\\ \cos \beta \sin \gamma & \cos \alpha \cos \gamma + \sin \alpha \sin \beta \sin \gamma & -\sin \alpha \cos \gamma + \cos \alpha \sin \beta \sin \gamma \\ -\sin \beta & \sin \alpha \cos \beta & \cos \alpha \cos \beta\end{matrix}\right]\]

where
\[\alpha, \beta, \gamma\]
are Euler angles (rotations about the X, Y, and Z axes of your coordinate system), respectively.

A multiplication of a 3-vector with such a matrix would yield a 3-vector rotated by the three angles. It follows, that for example to get the transformed x-component of the vector v [xyz], you would calculate

\[x' = x*cos \beta cos \gamma + y*cos \beta sin \gamma + z*-sin \beta \]

And do similarly for the y and z components with the second and third column of the matrix.

Someone please double check my math here, it's been a while since I've actually built a rotation matrix by hand :P

Answer 9

Thanx a lot for this great help.