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Chapter 10: Problem 41

If there is a \(60^{\circ}\) angle from the positive \(x\) -axis to the positive \(x^{\prime}\) -axis, explain how to obtain the rotation formulas for \(x\) and \(y\)

Short Answer

Expert verified
The rotation formulas for coordinates x and y when there's a \(60^{\circ}\) rotation from the positive \(x\) -axis to the positive \(x^{\prime}\) -axis are \(x^{\prime} = 1/2x - \sqrt{3}/2y\) and \(y^{\prime} = \sqrt{3}/2x + 1/2y\).

Step by step solution

01

Understand the rotation matrix

The rotation matrix for a counterclockwise rotation by an angle \(\theta\) is \(\begin{bmatrix} cos(\theta) & -sin(\theta) \ sin(\theta) & cos(\theta) \end{bmatrix}\). This matrix is derived from the unit circle and basic trigonometry.
02

Substitute the given rotation angle in the rotation matrix

Substitute the given angle of \(60^{\circ}\) into the rotation matrix. Note that this should be converted to radians because the trigonometric functions in the matrix are in terms of radians. So, \(60^{\circ}\) is equivalent to \(\pi/3\) radians. The substitution yields \(\begin{bmatrix} cos(\pi/3) & -sin(\pi/3) \ sin(\pi/3) & cos(\pi/3) \end{bmatrix} = \begin{bmatrix} 1/2 & -\sqrt{3}/2 \ \sqrt{3}/2 & 1/2 \end{bmatrix}\).
03

Apply the rotation matrix to the original coordinates

To find the new transformed coordinates \((x^{\prime}, y^{\prime})\), multiply the rotation matrix by the original coordinate vector \((x, y)\). This yields \(\begin{bmatrix} 1/2 & -\sqrt{3}/2 \ \sqrt{3}/2 & 1/2 \end{bmatrix} \times \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 1/2x - \sqrt{3}/2y \ \sqrt{3}/2x + 1/2y \end{bmatrix}\).

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Most popular questions from this chapter

In Exercises \(61-66,\) find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. $$ \left\\{\begin{aligned} 4 x^{2}+y^{2} &=4 \\ x+y &=3 \end{aligned}\right. $$

Describe a viewing rectangle, or window, such as [-30, 30, 3] by [-8, 4, 1], that shows a complete graph of each polar equation and minimizes unused portions of the screen. $$ r=\frac{8}{1+\cos \theta} $$

Describe a viewing rectangle, or window, such as [-30, 30, 3] by [-8, 4, 1], that shows a complete graph of each polar equation and minimizes unused portions of the screen. $$ r=\frac{8}{1-\cos \theta} $$

Describe a viewing rectangle, or window, such as [-30, 30, 3] by [-8, 4, 1], that shows a complete graph of each polar equation and minimizes unused portions of the screen. $$ r=\frac{16}{3+5 \cos \theta} $$

Write an equation for the path of each of the following elliptical orbits. Then use a graphing utility to graph the two ellipses in the same viewing rectangle. Can you see why early astronomers had difficulty detecting that these orbits are ellipses rather than circles? \(\cdot\) Earth's orbit: Length of major axis: 186 million miles Length of minor axis: 185.8 million miles \(\cdot\) Mars's orbit: Length of major axis: 283.5 million miles Length of minor axis: 278.5 million miles
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