Math Problem Statement
Solution
The problem is asking to rotate the linear function by around the origin (0,0). Let's solve it step by step.
Step 1: Parametrize the line equation
The line can be written as . We will rotate the points on this line around the origin.
Step 2: Rotation matrix
The general formula for rotating a point by an angle counterclockwise around the origin is given by the rotation matrix:
x' \\ y' \end{pmatrix} = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$$ For $$\theta = 60^\circ$$, the values are $$\cos 60^\circ = \frac{1}{2}$$ and $$\sin 60^\circ = \frac{\sqrt{3}}{2}$$. ### Step 3: Apply the rotation Substituting these values, the rotation matrix becomes: $$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix} \begin{pmatrix} x \\ x - 3 \end{pmatrix}$$ Now, perform the matrix multiplication to get the new coordinates $$(x', y')$$. ### Step 4: Simplify the new equation Performing the matrix multiplication: $$x' = \frac{1}{2}x - \frac{\sqrt{3}}{2}(x - 3) = \frac{1}{2}x - \frac{\sqrt{3}}{2}x + \frac{3\sqrt{3}}{2}$$ $$y' = \frac{\sqrt{3}}{2}x + \frac{1}{2}(x - 3) = \frac{\sqrt{3}}{2}x + \frac{1}{2}x - \frac{3}{2}$$ Now simplify these equations: $$x' = \left(\frac{1 - \sqrt{3}}{2}\right)x + \frac{3\sqrt{3}}{2}$$ $$y' = \left(\frac{\sqrt{3} + 1}{2}\right)x - \frac{3}{2}$$ This gives the parametric form of the line after rotation. Let me know if you'd like further details or clarifications on this process. ### Related questions: 1. How would you rotate the function if the angle were $$90^\circ$$ instead of $$60^\circ$$? 2. How can you express a general transformation of a linear function using matrices? 3. What is the geometric significance of rotating a line around the origin? 4. How would this process change if the line were rotated about a different point? 5. Can you find the intersection point of the original and rotated lines? ### Tip: When rotating a line, always use parametric equations for the original line to make the matrix multiplication straightforward.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Equations
Coordinate Geometry
Rotation Transformation
Formulas
Rotation matrix: [x', y'] = [cos(θ) -sin(θ); sin(θ) cos(θ)] * [x; y]
For θ = 60°, cos(60°) = 1/2 and sin(60°) = √3/2
Theorems
Rotation in Cartesian Plane
Suitable Grade Level
Grades 10-12
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