Given The Matrix:${ B = \left[\begin{array}{cc} -0.3090 & -0.9511 \ 0.9511 & -0.3090 \end{array}\right] }$Which Of The Following Angles Does The Matrix Represent?A. ${ 50^{\circ}\$} B. ${ 72^{\circ}\$} C.

Introduction

In linear algebra, a rotational matrix is a square matrix that represents a rotation in a two-dimensional or three-dimensional space. These matrices are used to describe the rotation of objects in various fields, including physics, engineering, and computer graphics. In this article, we will explore the concept of rotational matrices and how to determine the angle of rotation represented by a given matrix.

Rotational Matrix Formula

A rotational matrix in two dimensions can be represented by the following formula:

${ R(\theta) = \left[\begin{array}{cc} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{array}\right] }$

where $\theta$ is the angle of rotation in radians.

Given Matrix

We are given the following matrix:

${ B = \left[\begin{array}{cc} -0.3090 & -0.9511 \\ 0.9511 & -0.3090 \end{array}\right] }$

Our goal is to determine which angle this matrix represents.

Determining the Angle of Rotation

To find the angle of rotation, we can compare the given matrix with the rotational matrix formula. We can see that the elements of the given matrix match the formula, but with some differences in the signs.

Let's rewrite the given matrix as:

${ B = \left[\begin{array}{cc} -\cos(\theta) & -\sin(\theta) \\ \sin(\theta) & -\cos(\theta) \end{array}\right] }$

Comparing this with the rotational matrix formula, we can see that the angle of rotation is the same, but the signs of the elements are different.

Using Trigonometric Identities

We can use trigonometric identities to simplify the given matrix and find the angle of rotation.

Recall that $\cos(-\theta) = \cos(\theta)$ and $\sin(-\theta) = -\sin(\theta)$. We can rewrite the given matrix as:

${ B = \left[\begin{array}{cc} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{array}\right] }$

Now, we can see that the given matrix represents a rotation of $\theta$ radians.

Finding the Angle

To find the angle of rotation, we can use the fact that the matrix represents a rotation of $\theta$ radians. We can set up the following equation:

${ \left[\begin{array}{cc} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{array}\right] = \left[\begin{array}{cc} -0.3090 & -0.9511 \\ 0.9511 & -0.3090 \end{array}\right] }$

We can equate the corresponding elements of the two matrices and solve for $\theta$.

Solving for $\theta$

Equating the corresponding elements of the two matrices, we get:

${ \cos(\theta) = -0.3090 }$ ${ \sin(\theta) = -0.9511 }$

We can use the inverse tangent function to find the angle of rotation:

${ \theta = \arctan\left(\frac{\sin(\theta)}{\cos(\theta)}\right) }$

Substituting the values of $\sin(\theta)$ and $\cos(\theta)$, we get:

${ \theta = \arctan\left(\frac{-0.9511}{-0.3090}\right) }$

Evaluating the expression, we get:

${ \theta \approx 72^{\circ} }$

Conclusion

In this article, we explored the concept of rotational matrices and how to determine the angle of rotation represented by a given matrix. We used trigonometric identities to simplify the given matrix and find the angle of rotation. We found that the given matrix represents a rotation of approximately $72^{\circ}$.

References

• [1] "Linear Algebra and Its Applications" by Gilbert Strang
• [2] "Calculus: Early Transcendentals" by James Stewart
• [3] "Mathematics for Computer Graphics" by Michael E. Mortenson

Discussion

What are some other applications of rotational matrices in mathematics and computer science? How can we use rotational matrices to solve problems in physics and engineering? Share your thoughts and ideas in the comments below!

Introduction

In our previous article, we explored the concept of rotational matrices and how to determine the angle of rotation represented by a given matrix. In this article, we will answer some frequently asked questions about rotational matrices and provide additional insights into their applications.

Q: What is a rotational matrix?

A: A rotational matrix is a square matrix that represents a rotation in a two-dimensional or three-dimensional space. It is used to describe the rotation of objects in various fields, including physics, engineering, and computer graphics.

Q: How do I determine the angle of rotation represented by a given matrix?

A: To determine the angle of rotation, you can compare the given matrix with the rotational matrix formula. You can also use trigonometric identities to simplify the matrix and find the angle of rotation.

Q: What are some common applications of rotational matrices?

A: Rotational matrices have many applications in mathematics, physics, engineering, and computer science. Some common applications include:

• Computer graphics: Rotational matrices are used to rotate objects in 3D space, creating realistic animations and simulations.
• Physics: Rotational matrices are used to describe the rotation of objects in physics, including the rotation of planets and stars.
• Engineering: Rotational matrices are used to design and analyze mechanical systems, including gears and other rotating components.
• Mathematics: Rotational matrices are used to study the properties of rotations and to develop new mathematical theories.

Q: How do I create a rotational matrix?

A: To create a rotational matrix, you can use the following formula:

${ R(\theta) = \left[\begin{array}{cc} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{array}\right] }$

where $\theta$ is the angle of rotation in radians.

Q: What are some common mistakes to avoid when working with rotational matrices?

A: Some common mistakes to avoid when working with rotational matrices include:

• Incorrectly calculating the angle of rotation: Make sure to use the correct formula and trigonometric identities to calculate the angle of rotation.
• Using the wrong matrix: Make sure to use the correct rotational matrix formula and not a different type of matrix.
• Not considering the sign of the angle: Make sure to consider the sign of the angle when working with rotational matrices.

Q: How do I use rotational matrices in computer graphics?

A: To use rotational matrices in computer graphics, you can follow these steps:

1. Define the rotation axis: Define the axis of rotation for the object.
2. Calculate the rotation angle: Calculate the angle of rotation using the rotational matrix formula.
3. Create the rotational matrix: Create the rotational matrix using the calculated angle and rotation axis.
4. Apply the rotation: Apply the rotation to the object using the rotational matrix.

Q: What are some advanced topics in rotational matrices?

A: Some advanced topics in rotational matrices include:

• Quaternions: Quaternions are a way to represent 3D rotations using a four-dimensional vector.
• Euler angles: Euler angles are a way to represent 3D rotations using three angles.
• Rotation groups: Rotation groups are a way to study the properties of rotations and to develop new mathematical theories.

Conclusion

In this article, we answered some frequently asked questions about rotational matrices and provided additional insights into their applications. We hope this article has been helpful in understanding the concept of rotational matrices and how to use them in various fields.

References

• [1] "Linear Algebra and Its Applications" by Gilbert Strang
• [2] "Calculus: Early Transcendentals" by James Stewart
• [3] "Mathematics for Computer Graphics" by Michael E. Mortenson

Discussion

What are some other applications of rotational matrices in mathematics and computer science? How can we use rotational matrices to solve problems in physics and engineering? Share your thoughts and ideas in the comments below!