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📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 \mathbf{p} \cdot \mathbf{q} = 4(2) + (-1)(0) + 3(-1) = 8 + 0 - 3 = 5

📌 Next, find $\|\mathbf{q}\|^2$:

📌 \|\mathbf{q}\|^2 = 2^2 + 0^2 + (-1)^2 = 4 + 0 + 1 = 5

📌 Now compute the scalar multiplier:

📌 \frac{5}{5} = 1

📌 \text{proj}_{\mathbf{q}} \mathbf{p} = 1 \cdot \begin{pmatrix} 2 \\ 0 \\ -1 \end{pmatrix} = \begin{pmatrix} 2 \\ 0 \\ -1 \end{pmatrix}

📌 \boxed{\begin{pmatrix} 2 \\ 0 \\ -1 \end{pmatrix}}

📌 An astrophysics researcher analyzes stellar motion with vectors $\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$ and $\mathbf{s} = \begin{pmatrix} -2 \\ 1 \\ 4 \end{pmatrix}$. Find the area of the parallelogram spanned by $\mathbf{r}$ and $\mathbf{s}$.

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