Image of vectors when rotated

smeshy · #1 $Old$ November 2nd, 2009, 03:26 PM

Hi, I need some help with a Maths homework problem:

Find the image of the vectors (3,0) and (3,4) when rotated counter-clockwise by pi/4.

I'm not too sure how to start it. Do I use pi/4 as the angle between the vector and it's image?

earboth · #2 $Old$ November 3rd, 2009, 12:52 AM

Quote:

Originally Posted by smeshy $View Post$

Hi, I need some help with a Maths homework problem:

Find the image of the vectors (3,0) and (3,4) when rotated counter-clockwise by pi/4.

I'm not too sure how to start it. Do I use pi/4 as the angle between the vector and it's image?

1. I assume that the origin is the center of rotation.

2. Let (X, Y) denote the image of (x,y) by rotation around the origin. With you question the angle of rotation is $\alpha = \frac \pi4$

3. The rotation is described by a system of equations:

$\begin{array}{l}X=x\cdot \cos(\alpha) - y \cdot \sin(\alpha) \\ Y=x\cdot \sin(\alpha) + y \cdot \cos(\alpha) \end{array}$

4. Plug in the coordinates of the original points to get the coordinates of the image.