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Home » Factor numerator: \( rac(x - 2)(x + 2)x - 2 \). - ClickBalance

Factor numerator: \( rac(x - 2)(x + 2)x - 2 \). - ClickBalance

📅 May 17, 2026 👤 scraface

May 17, 2026

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📌 \boxed{\begin{bmatrix} 4 \\ -3 \end{bmatrix} \text{ or } \begin{bmatrix} -4 \\ 3 \end{bmatrix}}

📌 Question:** An angel investor is analyzing the growth patterns of a biotechnology startup. The growth rate of their revenue, modeled as a vector \(\mathbf{r} = \begin{bmatrix} 2 \\ 3 \\ 6 \end{bmatrix}\), needs to be decomposed into components parallel and perpendicular to the vector \(\mathbf{a} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}\). Find the component of \(\mathbf{r}\) that is parallel to \(\mathbf{a}\).

📌 Solution:** The component of \(\mathbf{r}\) parallel to \(\mathbf{a}\) is given by the projection of \(\mathbf{r}\) onto \(\mathbf{a}\):

📌 \text{proj}_{\mathbf{a}} \mathbf{r} = \frac{\mathbf{r} \cdot \mathbf{a}}{\mathbf{a} \cdot \mathbf{a}} \mathbf{a}

📌 \mathbf{r} \cdot \mathbf{a} = 2 \cdot 1 + 3 \cdot 1 + 6 \cdot 1 = 11

📌 \mathbf{a} \cdot \mathbf{a} = 1^2 + 1^2 + 1^2 = 3

📌 Thus, the projection is:

📌 \text{proj}_{\mathbf{a}} \mathbf{r} = \frac{11}{3} \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} \frac{11}{3} \\ \frac{11}{3} \\ \frac{11}{3} \end{bmatrix}

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