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Home » Solution: First, find the constant $ k $ using the given values: - AMAZONAWS

Solution: First, find the constant $ k $ using the given values: - AMAZONAWS

📅 March 5, 2026 👤 scraface

Mar 05, 2026

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📌 \end{vmatrix} = \langle 3v_2 - 2v_3, 3v_1 - v_3, 2v_1 - v_2 \rangle.

📌 Set this equal to $\mathbf{c} - \mathbf{b} = \langle 3, 3, 3 \rangle$:

📌 3v_2 - 2v_3 = 3 \\

📌 3v_1 - v_3 = 3 \\

📌 2v_1 - v_2 = 3

📌 Solve the system: From the third equation, $v_2 = 2v_1 - 3$. Substitute into the first equation: $3(2v_1 - 3) - 2v_3 = 3 \Rightarrow 6v_1 - 9 - 2v_3 = 3 \Rightarrow 6v_1 - 2v_3 = 12$. From the second equation, $v_3 = 3v_1 - 3$. Substitute into $6v_1 - 2(3v_1 - 3) = 12 \Rightarrow 6v_1 - 6v_1 + 6 = 12 \Rightarrow 6 = 12$, which is impossible. Thus, no solution exists.

📌 Question: A glaciologist models ice flow with vectors $\mathbf{u}$ and $\mathbf{w}$, where $\|\mathbf{u}\| = 5$, $\|\mathbf{w}\| = 7$, and the angle between them is $60^\circ$. Compute the magnitude of $\mathbf{u} \times \mathbf{w}$.

📌 Solution: The magnitude of the cross product is given by $\|\mathbf{u} \times \mathbf{w}\| = \|\mathbf{u}\| \|\mathbf{w}\| \sin \theta$. Here, $\theta = 60^\circ$,

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