Linear transformation rotation

In summary, the given transformation involves reflecting points through -3pi/4 radians clockwise and then through the horizontal x1-axis. The resulting points are (-1/sqrt2, 1/sqrt2) due to the combination of rotation and reflection. The confusion about the use of sqrt2 on the bottom can be clarified by understanding that 2 is equal to (sqrt(2))*(sqrt(2)).
  • #1
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Homework Statement


T: R2-->R2 first reflects points through -3pi/4 radian (clockwise) and then reflects points through the horizontal x1-axis. [Hint T(e1)= (-1/sqrt2, 1/sqrt2)



The Attempt at a Solution


I just don't understand why the points would be (-1/sqrt2, 1/sqrt2). If it's -3pi/4, why wouldn't it be (-(sqrt2)/2, -(sqrt2)/2)?
 
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  • #2
Did you miss the part about the reflection or are you rotating the wrong way? Rotating (1, 0) -3pi/4 radians (clockwise as it says) gives you (-1/sqrt(2), -1/sqrt(2)) but then the reflection through the horizontal axis changes that to (-1/sqrt(2),+1/sqrt(2)).
 
  • #3
Doesn't -3pi/4 radians correlate to (-(sqrt2)/2, -(sqrt2)/2) though (on a unit circle)? Where is the sqrt2 on the bottom coming from?
 
  • #4
Are you aware that 2= (sqrt(2))*(sqrt(2))? 1/sqrt(2) is exactly the same as sqrt(2)/2.
 

Related to Linear transformation rotation

1. What is a linear transformation rotation?

A linear transformation rotation is a mathematical operation that maps points from one coordinate system to another by rotating them around a fixed point. It can be described as a combination of a translation and a rotation.

2. How is a linear transformation rotation represented?

In two-dimensional space, a linear transformation rotation can be represented by a 2x2 matrix. In three-dimensional space, it can be represented by a 3x3 matrix. The matrix contains the values for the rotation and translation components of the transformation.

3. What is the difference between a linear transformation rotation and a reflection?

A linear transformation rotation involves rotating points around a fixed point, while a reflection involves flipping points across a line. Both operations can be represented by matrices, but the values in the matrices will be different.

4. How does a linear transformation rotation affect the shape of an object?

A linear transformation rotation can change the orientation, size, and position of an object. It can also alter the shape of an object by stretching or shrinking it in certain directions. The amount of change depends on the values in the transformation matrix.

5. What are some real-world applications of linear transformation rotations?

Linear transformation rotations have many practical applications, including computer graphics, robotics, and physics. They are used to manipulate and animate 3D objects in video games and movies, to control the movement of robots and drones, and to model rotational motion in physics simulations.

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