Solving Reflection Matrix Homework: Angle & Rotation About Origin

In summary, the conversation discusses determining whether a given matrix is a rotation about the origin or a reflection about the positive x axis, as well as finding the angle between the line and the x-axis. The conversation concludes that the matrix is both a rotation and a reflection, with an angle of pi/3. The formula for rotation through an angle is also mentioned.
  • #1
Lchan1
39
0

Homework Statement


Given the matrix A=
-1/2 root(3)/2
root(3)/2 1/2

determine if the matrix is a rotation about the origin or a reflection about the positive x axis
and find the angle between the that the line makes with the x-axis



Homework Equations





The Attempt at a Solution



I found the det(A)=-1 can I concluded that the matrix reflects vectors?
and also how do you find the angle?
I tried doing projection onto the axis. but didnt get me anywhere
please help..
 
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  • #2
Hi Lchan1! :smile:

(have a square-root: √ :wink:)
Lchan1 said:
I found the det(A)=-1 can I concluded that the matrix reflects vectors?

Yup! :smile:
and also how do you find the angle?

learn this:

rotation through angle θ is

cosθ sinθ
-sinθ cosθ :wink:
 
  • #3
does that mean it's a rotation and a reflection?
pi/3 will be the angle?
 
  • #4
Lchan1 said:
does that mean it's a rotation and a reflection?
pi/3 will be the angle?

sorry, I was a bit too cryptic last time :redface:

a combination of a rotation and a reflection is a reflection …

it would have been better if I'd said

rotation through angle θ is

cosθ sinθ
-sinθ cosθ

reflection about y-axis is

-1 0
0 1

and if you combine them you get … ? :smile:
 

1. What is a reflection matrix?

A reflection matrix is a mathematical representation of a reflection transformation in a coordinate plane. It is a 2x2 matrix that represents the change in coordinates when an object is reflected over a line or plane.

2. How do you solve for the reflection matrix?

The reflection matrix can be solved by following the steps of a reflection transformation. First, determine the line or plane of reflection. Then, create a coordinate grid and plot the original points. Next, apply the reflection transformation to each point and plot the new points. Finally, use the coordinates of the original and new points to create a 2x2 matrix.

3. What is the difference between angle and rotation about origin?

Angle and rotation about origin are both measures of the amount and direction of rotation, but they differ in their points of reference. Angle measures the rotation of an object in relation to a fixed point, while rotation about origin measures the rotation of an object in relation to its own center.

4. How does rotation about origin affect coordinates?

Rotation about origin changes the coordinates of an object by applying a rotation transformation. The new coordinates of each point are calculated by using the rotation formula, which involves the angle of rotation and the original coordinates of the point.

5. Can you provide an example of solving reflection matrix homework?

Yes, for example, if we have a point (3,4) and we want to reflect it over the y-axis, we would first plot the point on a coordinate grid. Then, we would apply the reflection transformation by swapping the x-coordinate with its opposite, resulting in (-3,4). Finally, we would use the original and new coordinates to create the reflection matrix: [(-1 0), (0 1)].

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