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Home » \[ 1 - t^2 = 0 \] - Groen Casting

\[ 1 - t^2 = 0 \] - Groen Casting

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Take derivative and set to 0:

📌 \frac{d}{dt}(6t^2 + 4t + 30) = 12t + 4 = 0 \Rightarrow t = -\frac{1}{3}

📌 Substitute $t = -\frac{1}{3}$ into $\vec{r}(t)$:

📌 \vec{r}\left(-\frac{1}{3}\right) = \begin{pmatrix} 1 + 2(-\frac{1}{3}) \\ -\frac{1}{3} \\ -3 - (-\frac{1}{3}) \end{pmatrix} = \begin{pmatrix} 1 - \frac{2}{3} \\ -\frac{1}{3} \\ -3 + \frac{1}{3} \end{pmatrix} = \begin{pmatrix} \frac{1}{3} \\ -\frac{1}{3} \\ -\frac{8}{3} \end{pmatrix}

📌 Thus, the closest point is:

📌 \boxed{\begin{pmatrix} \frac{1}{3} \\ -\frac{1}{3} \\ -\frac{8}{3} \end{pmatrix}}

📌 Question: Find the value of $x$ such that the vectors $\begin{pmatrix} 2 \\ x \\ -3 \end{pmatrix}$ and $\begin{pmatrix} 4 \\ -1 \\ 1 \end{pmatrix}$ are orthogonal.

📌 Two vectors are orthogonal if their dot product is zero. Compute the dot product:

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