# Rotation of Vectors: Comparing Matrices

• ahmed markhoos
In summary, the rotation of vectors refers to the transformation of a vector by rotating it around a given point or axis. This transformation is represented mathematically by a rotation matrix, which is a square matrix with values corresponding to the cosine and sine of the rotation angle. To compare rotation matrices, you can look at their values and determine if they represent the same rotation or different rotations. This comparison is significant because it helps us understand how vectors are transformed in different spaces and determine if two rotations are equivalent or different, which is crucial in various scientific and mathematical applications.
ahmed markhoos
< Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown >

Hello,

I have a problem with rotation matrices, its just a comparison problem. Θ must be 240 or -120, I don't know how the book show the answer like that, these are the two matrices

\begin{array}--1/2&\sqrt{(3)}/2\\-\sqrt{(3)}/2&-1/2\end{array}

with

\begin{array}ccosΘ&-sinΘ\\sinΘ&cosΘ\end{array}

- I tried to take element 1,1 and 2,1 , it gives 120 & -60. How is that suppose to be 240 or -120?

Last edited by a moderator:
When you match the 11 and 21 components, you get two equations. Each of these equations has two solutions in the interval [0,360). Only one of the two solutions to the first equation will be a solution to the second equation as well.

ahmed markhoos
ahmed markhoos said:
< Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown >

Hello,

I have a problem with rotation matrices, its just a comparison problem. Θ must be 240 or -120, I don't know how the book show the answer like that, these are the two matrices

\begin{array}--1/2&\sqrt{(3)}/2\\-\sqrt{(3)}/2&-1/2\end{array}

with

\begin{array}ccosΘ&-sinΘ\\sinΘ&cosΘ\end{array}

- I tried to take element 1,1 and 2,1 , it gives 120 & -60. How is that suppose to be 240 or -120?

Look at the first column of your matrix: you have ##\cos(\theta) = -1/2## and ##\sin(\theta) = -\sqrt{3}/2##. Since both ##\sin(\theta)## and ##\cos(\theta)## are ##< 0##, in what quadrant must ##\theta## lie? Just draw a picture of you need more insight.

-120 and 240 is the same angle.

Nitpick: They're equivalent, but not the same. (cos 240,sin 240) and (cos(-120),sin(-120)) are however the same point in ##\mathbb R^2##.

theodoros.mihos

## 1. What is meant by "rotation of vectors"?

The rotation of vectors refers to the transformation of a vector by rotating it around a given point or axis. This can be done in two-dimensional space by changing the direction of the vector or in three-dimensional space by changing both the direction and magnitude of the vector.

## 2. How is the rotation of a vector represented mathematically?

In mathematics, the rotation of a vector is represented by a matrix. This matrix is typically a square matrix with values that correspond to the cosine and sine of the rotation angle. The resulting rotated vector can be calculated by multiplying the original vector by this rotation matrix.

## 3. What is a rotation matrix?

A rotation matrix is a square matrix that is used to rotate a vector in a given space. It is a mathematical representation of the transformation that occurs when a vector is rotated around a point or axis. A rotation matrix typically has 3 rows and 3 columns, and its values can be calculated using trigonometric functions.

## 4. How do you compare rotation matrices?

To compare rotation matrices, you can look at their values and determine if they represent the same rotation or different rotations. If the matrices have the same values, they represent the same rotation. Additionally, you can multiply the matrices and if the result is an identity matrix, then the two matrices represent the same rotation.

## 5. What is the significance of comparing rotation matrices?

Comparing rotation matrices is important because it allows us to understand how vectors are transformed in different spaces. It also helps us to determine if two rotations are equivalent or different. This is crucial in many scientific and mathematical applications, such as computer graphics and robotics.

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