# How to solve the following matrix rotation

• MHB
• Logan Land
In summary: In this case, you could use the Gram-Schmidt orthonormalization procedure to obtain another basis. What would be the change of basis matrix?
Logan Land
Find the standard matrix for the rotation of 60◦ about the axis determined by the vector v = (3, 4, 5).

do I multiply each x,y,z by pi/3?

doesnt seem like it should be that simple

ok so I figured out that there is a rotational matrix in basic form that is

cos -sin
sin cos

so a rotation by 60degrees gives me a rotational matrix of
1/2 -sqrt3/2
sqrt3/2 1/2

now from here where would I proceed multiply it by v=(3,4,5)?

The matrix you gave is for 2D vectors. Here is a hint: find a transformation that sends (3, 4, 5) to some standard axis (say, the x-axis, i.e. to a vector of the form (t, 0, 0)) then rotate the x-axis, or whichever axis you picked, 60 degrees (that can be done using the 2D matrix you gave: for standard axes, the 3D rotation matrix is similar to the 2D one, as one axis is left fixed). Then transform back. Can you come up with a basis that has (3, 4, 5) as a basis vector? What would be the change of basis matrix? Hence, what would the rotation matrix be?

Bacterius said:
The matrix you gave is for 2D vectors. Here is a hint: find a transformation that sends (3, 4, 5) to some standard axis (say, the x-axis, i.e. to a vector of the form (t, 0, 0)) then rotate the x-axis, or whichever axis you picked, 60 degrees (that can be done using the 2D matrix you gave: for standard axes, the 3D rotation matrix is similar to the 2D one, as one axis is left fixed). Then transform back. Can you come up with a basis that has (3, 4, 5) as a basis vector? What would be the change of basis matrix? Hence, what would the rotation matrix be?

im confused
i don't no how to get 3,4,5 to t,0,0 or 0,t,0 or 0,0,t
could i get an example then i can try and figure out my problem

can i use a rotational matrix that's 3d? and not have to perform a transformation?

LLand314 said:
im confused
i don't no how to get 3,4,5 to t,0,0 or 0,t,0 or 0,0,t
could i get an example then i can try and figure out my problem

can i use a rotational matrix that's 3d? and not have to perform a transformation?

Have you learned change of basis matrices? What are you expected to use to solve this problem?

Bacterius said:
Have you learned change of basis matrices? What are you expected to use to solve this problem?

we only have 1 vector set though... (3,4,5)
what would I use to obtain another basis?
We've always been given multiple vectors

## 1. What is matrix rotation?

Matrix rotation is a mathematical operation that involves changing the orientation of a matrix by rotating it around its center or a specified point.

## 2. How do I determine the direction of rotation?

The direction of rotation is determined by the axis of rotation. If the axis of rotation is the x-axis, then the rotation will be clockwise if viewed from the positive y-axis and counterclockwise if viewed from the negative y-axis. Similarly, if the axis of rotation is the y-axis, the rotation will be clockwise if viewed from the positive x-axis and counterclockwise if viewed from the negative x-axis.

## 3. What are the steps for rotating a matrix?

The steps for rotating a matrix are as follows:

1. Identify the axis of rotation and determine the direction of rotation.
2. Find the coordinates of the center or the specified point of rotation.
3. Subtract the coordinates of the center from each point in the matrix to translate the matrix so that the center becomes the origin.
4. Apply the rotation formula to each point in the translated matrix.
5. Add the coordinates of the center back to each point to translate the matrix back to its original position.

## 4. What is the formula for rotating a matrix?

The formula for rotating a point (x,y) around the origin (0,0) by an angle θ is:

x' = xcosθ - ysinθ

y' = xsinθ + ycosθ

Where x' and y' are the new coordinates after rotation.

## 5. Can a matrix be rotated by any angle?

Yes, a matrix can be rotated by any angle. The rotation formula can be modified to accommodate any angle of rotation. However, keep in mind that rotating by certain angles may result in a distorted or skewed matrix.

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