# Two dimensional rotational Matrix

• MHB
• cbarker1
In summary: Sorry about that.In summary, the conversation discusses the question from Problem 1.8 in the textbook Introduction to Electrodynamics and how to prove that the two-dimensional rotation matrix preserves the length of vector A. The conversation includes a vector equation and instructions on how to expand it to prove the statement. The speaker clarifies that they meant to say "preserve" instead of "perverse".
cbarker1
Gold Member
MHB
Dear Everybody,

I am trying to learn about the electrodynamics. I am using the textbook, Introduction to Electrodynamics (2nd Ed) by D. J. Griffiths. I am working on the Problem 1.8. The question state:

Prove that the two-dimensional rotation matrix perverse the length of A. (That is, show that $(A_y')^2+(A_z')^2=(A_y)^2+(A_z)^2$)

$\left(\begin{array}{cc} A_y'\\ A_z'\end{array}\right)$=$\left(\begin{array}{cc} cos(\phi) & sin(\phi)\\ -sin(\phi) & cos(\phi) \end{array} \right)$ $\left(\begin{array}{cc} A_y\\A_z\end{array}\right)$

I am struck at the beginning.

Cbarker1 said:
Dear Everybody,

I am trying to learn about the electrodynamics. I am using the textbook, Introduction to Electrodynamics (2nd Ed) by D. J. Griffiths. I am working on the Problem 1.8. The question state:

Prove that the two-dimensional rotation matrix perverse the length of A. (That is, show that $(A_y')^2+(A_z')^2=(A_y)^2+(A_z)^2$)

$\left(\begin{array}{cc} A_y'\\ A_z'\end{array}\right)$=$\left(\begin{array}{cc} cos(\phi) & sin(\phi)\\ -sin(\phi) & cos(\phi) \end{array} \right)$ $\left(\begin{array}{cc} A_y\\A_z\end{array}\right)$

I am struck at the beginning.
From the vector equation you have
$$\displaystyle A_{y'} = cos( \theta ) A_y + sin( \theta ) A_z$$
$$\displaystyle A_{z'} = -sin( \theta ) A_y + cos( \theta ) A_z$$

So
$$\displaystyle (A_{y'})^2 + (A_{z'})^2 = \left ( cos( \theta ) A_y + sin( \theta ) A_z \right ) ^2 + \left ( -sin( \theta ) A_y + cos( \theta ) A_z \right ) ^2$$

Expand it out (it's not quite as bad as it looks) and you will get
$$\displaystyle (A_{y'})^2 + (A_{z'})^2 = \left ( sin^2 ( \theta ) + cos^2( \theta ) \right ) (A_{y})^2 + \left ( sin^2( \theta ) + cos^2 ( \theta ) \right ) (A_z)^2$$

(Hint: The terms in $$\displaystyle A_y ~ A_z$$ cancel out.)

Can you finish out the details?

-Dan

I hope you mean "preserve", not "perverse"!

Country Boy said:
I hope you mean "preserve", not "perverse"!
I meant to preserve.

## 1. What is a two-dimensional rotational matrix?

A two-dimensional rotational matrix is a mathematical representation of a rotation in a two-dimensional coordinate system. It is a square matrix that contains elements which determine the angle and direction of the rotation.

## 2. How is a two-dimensional rotational matrix used?

A two-dimensional rotational matrix is used to rotate points or objects in a two-dimensional plane. It is commonly used in computer graphics, physics, and engineering applications to manipulate and transform objects.

## 3. How can I create a two-dimensional rotational matrix?

To create a two-dimensional rotational matrix, you need to know the angle of rotation and the direction of rotation. You can use trigonometric functions to determine the values for the elements of the matrix. The resulting matrix will have the form of [cosθ -sinθ; sinθ cosθ], where θ is the angle of rotation.

## 4. What is the difference between a two-dimensional rotational matrix and a two-dimensional translation matrix?

A two-dimensional rotational matrix rotates points or objects in a coordinate system, while a two-dimensional translation matrix moves them. A rotation matrix changes the orientation of an object, while a translation matrix changes its position in the coordinate plane.

## 5. Can a two-dimensional rotational matrix be used to rotate objects in three-dimensional space?

No, a two-dimensional rotational matrix can only be used to rotate points or objects in a two-dimensional plane. To rotate objects in three-dimensional space, a three-dimensional rotational matrix is needed, which has additional elements to account for the third dimension.

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