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These 5 Hidden Knobs Are Transforming Everyday Objects (Shop Now!) - Abbey Badges

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 If \(x = 4\), then \(y = -\frac{3}{4}(4) = -3\). If \(x = -4\), then \(y = -\frac{3}{4}(-4) = 3\).

📌 Thus, the possible vectors are \(\mathbf{v} = \begin{bmatrix} 4 \\ -3 \end{bmatrix}\) or \(\mathbf{v} = \begin{bmatrix} -4 \\ 3 \end{bmatrix}\).

📌 \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

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