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Chapter 6: Problem 56

Find the matrix that produces the rotation. \(60^{\circ}\) about the \(x\) -axis

Short Answer

Expert verified
The rotation matrix for a \(60^{\circ}\) rotation about the x-axis is \[\begin{bmatrix} 1 & 0 & 0 \\ 0 & \frac{1}{2} & -\frac{\sqrt{3}}{2} \\ 0 & \frac{\sqrt{3}}{2} & \frac{1}{2} \end{bmatrix}\]

Step by step solution

01

Formulate the Formula

The general matrix formula for rotation about the x-axis by an angle \( \theta \) is: \[\begin{bmatrix} 1 & 0 & 0 \\ 0 & cos(\theta) & -sin(\theta) \\ 0 & sin(\theta) & cos(\theta) \end{bmatrix}\]. Now, substitute \( \theta \) with \(60^\circ \)
02

Convert Degrees into Radians

To apply the trigonometric functions, convert the angle from degrees into radians. To do this, use the formula \( radians = degrees \times \frac{\pi}{180} \). So, \(60^{\circ} = \frac{\pi}{3}
03

Substitute the Angle into the Matrix

Now, substitute \( \theta = \frac{\pi}{3} \) into the matrix formula and compute each trigonometric value to obtain the rotation matrix. So our rotation matrix could be represented as: \[\begin{bmatrix} 1 & 0 & 0 \\ 0 & cos(\frac{\pi}{3}) & -sin(\frac{\pi}{3}) \\ 0 & sin(\frac{\pi}{3}) & cos(\frac{\pi}{3}) \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \frac{1}{2} & -\frac{\sqrt{3}}{2} \\ 0 & \frac{\sqrt{3}}{2} & \frac{1}{2} \end{bmatrix}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Trigonometric Functions
Trigonometric functions are fundamental mathematical functions that relate the angles of a triangle to the lengths of its sides. They are based on the relationships defined within a unit circle. Here are the main trigonometric functions relevant for rotations:
  • Sine (\( \sin \)): This function gives the vertical component of an angle in a right triangle or on a unit circle. For example, \( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} \).

  • Cosine (\( \cos \)): This function gives the horizontal component of an angle in a right triangle or on a unit circle. For \( \cos(\frac{\pi}{3}) = \frac{1}{2} \).

  • Tangent (\( \tan \)): Though not directly used in 3D rotation matrices, it is the ratio of sine to cosine, or: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).

These functions are vital when converting geometric problems, like rotations, into algebraic equations, which are then solvable using a rotation matrix.
3D Rotation: An Insight
3D rotation is a transformation used to pivot an object around an axis in three-dimensional space. This is crucial in many fields such as computer graphics, robotics, and animation.
  • X-, Y-, and Z-Axes: Any object in 3D space can be rotated around these three perpendicular axes. In this exercise, the rotation occurs around the x-axis.

  • Specifying Rotation: The rotation is usually defined by an angle \( \theta \), represented in either degrees or radians. The angle determines how far the object pivots around the axis.

  • Applications: It is applied in simulations and modeling of real-world objects, allowing for realistic and dynamic views of three-dimensional objects.

Grasping 3D rotation helps in understanding complex spatial transformations essential for advanced geometric calculations.
Matrix Transformation Explained
Matrix transformations are mathematical operations that modify vectors, points, or other matrices by multiplying them with specific matrices to achieve a desired effect such as rotation, scaling, or translation.
  • Rotation Matrix: For a 3D rotation about the x-axis, a rotation matrix is utilized. It typically looks like:\[\begin{bmatrix} 1 & 0 & 0 \0 & \cos(\theta) & -\sin(\theta) \0 & \sin(\theta) & \cos(\theta) \end{bmatrix}\]The values inside are derived from trigonometric functions based on the rotation angle \( \theta \).

  • Transformation Process: Applying the matrix to a vector modifies its position. The matrix transforms by redistributing the values inside it, specifically using trigonometric evaluations like \( \sin \) and \( \cos \).

  • Utility: This provides a reliable method for rotating objects in 3D space efficiently, which can then be used in simulations and real-time applications.

By rotating matrices, complex spatial patterns and manipulations become manageable, aiding in both foundational learning and advanced problem-solving in mathematics and computer science.

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Most popular questions from this chapter

Sketch the image of the unit square [a square with vertices at \((0,0),(1,0),(1,1), \text { and }(0,1)]\) under the specified transformation. \(T\) is the shear represented by \(T(x, y)=(x+2 y, y)\)

find the matrix \(A^{\prime}\) for \(T\) relative to the basis \(B^{\prime}\) $$ \begin{aligned} &T: R^{2} \rightarrow R^{2}, T(x, y)=(5 x+4 y, 4 x+5 y)\\\ &B^{\prime}=\\{(12,-13),(13,-12)\\} \end{aligned} $$

Proof Prove Property 3 of Theorem 6.13: For square matrices \(A, B,\) and \(C\) of order \(n,\) if \(A\) is similar to \(B\) and \(B\) is similar to \(C,\) then \(A\) is similar to \(C\)

A translation in \(R^{2}\) is a function of the form \(T(x, y)=(x-h, y-k),\) where at least one of the constants \(h\) and \(k\) is nonzero. (a) Show that a translation in \(R^{2}\) is not a linear transformation. (b) For the translation \(T(x, y)=(x-2, y+1)\) determine the images of \((0,0),(2,-1),\) and \((5,4)\) (c) Show that a translation in \(R^{2}\) has no fixed points.

Proof Prove that if \(A\) is an idempotent matrix and \(B\) is similar to \(A,\) then \(B\) is idempotent. (Recall that an \(n \times n\) matrix \(A\) is idempotent when \(A=A^{2}\).)
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