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Trigonometry – Wind Turbine Blade Angle - Londonproperty

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 Set equal: $ 2x - 5 = -\frac{1}{2}x + 5 $ → $ \frac{5}{2}x = 10 $ → $ x = 4 $. Substitute $ x = 4 $ into $ y = 2x - 5 $: $ y = 3 $. The closest point is $ (4, 3) $, which coincides with the given point (implying the point lies on the line). However, verifying: $ 3 = -\frac{1}{2}(4) + 5 = -2 + 5 = 3 $. Thus, the closest point is $ \boxed{(4, 3)} $.

📌 Question: A data scientist models urban traffic flow with a transformation matrix $ \mathbf{M} $ that rotates traffic patterns by $ 45^\circ $ and scales them by $ \sqrt{2} $. Find $ \mathbf{M} $.

📌 Solution: A rotation matrix for $ \theta = 45^\circ $ is $ \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} = \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix} $. Scaling by $ \sqrt{2} $ multiplies each entry:

📌 \mathbf{M} = \sqrt{2} \cdot \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix} = \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix}.

📌 Thus, $ \boxed{\begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix}} $.

📌 Question: Planetary exploration engineers use a matrix $ \mathbf{M} $ to transform satellite coordinates. If $ \mathbf{M} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 3 \\ -2 \end{bmatrix} $ and $ \mathbf{M} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 4 \end{bmatrix} $, find $ \mathbf{M} $.

📌 Solution: The columns of $ \mathbf{M} $ are the images of the standard basis vectors. Thus, $ \mathbf{M} = \begin{bmatrix} 3 & 1 \\ -2 & 4 \end{bmatrix} $. Verification:

📌 \mathbf{M} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = 3 \cdot \begin{bmatrix} 1 \\ 0 \end{bmatrix} + 1 \cdot \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\ -2 \end{bmatrix}, \quad \mathbf{M} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = 1 \cdot \begin{bmatrix} 1 \\ 0 \end{bmatrix} + 4 \cdot \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 4 \end{bmatrix}.

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