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Home » \( d = \frac9545 = \frac199 \approx 2.11 \) - Silent Sales Machine

\( d = \frac9545 = \frac199 \approx 2.11 \) - Silent Sales Machine

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 3x + 4(-6) = 0

📌 3x - 24 = 0

📌 3x = 24 \quad \Rightarrow \quad x = 8

📌 Thus, the value of \(x\) that makes the vectors orthogonal is \(oxed{8}\).

📌 A civil engineer at the World Bank is designing a sustainable pedestrian bridge that follows a linear path defined by the parametric equation \(\mathbf{r}(t) = egin{pmatrix} 1 \ -2 \end{pmatrix} + t egin{pmatrix} 3 \ 1 \end{pmatrix}\). They need to locate the point on this line closest to a proposed water source located at \(egin{pmatrix} 5 \ 4 \end{pmatrix}\). Find the coordinates of this closest point.

📌 The closest point on a line to a given point is the orthogonal projection of that point onto the line.

📌 A point on the line: \(\mathbf{r}_0 = egin{pmatrix} 1 \ -2 \end{pmatrix}\)

📌 Direction vector: \(\mathbf{d} = egin{pmatrix} 3 \ 1 \end{pmatrix}\)

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