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Home » \[ a(3)^2 + b(3) + c = 5 \] - AdVision eCommerce

\[ a(3)^2 + b(3) + c = 5 \] - AdVision eCommerce

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 Solution: The angle $\theta$ between $\mathbf{a}$ and $\mathbf{b}$ is given by $\cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \|\mathbf{b}\|}$. Compute the dot product: $2(1) + (-3)(4) = 2 - 12 = -10$. Compute magnitudes: $\|\mathbf{a}\| = \sqrt{2^2 + (-3)^2} = \sqrt{13}$, $\|\mathbf{b}\| = \sqrt{1^2 + 4^2} = \sqrt{17}$. Thus, $\cos\theta = \frac{-10}{\sqrt{13}\sqrt{17}}$. Rationalizing, $\theta = \arccos\left(-\frac{10}{\sqrt{221}}\right)$. $\boxed{\arccos\left(-\dfrac{10}{\sqrt{221}}\right)}$

📌 Question: If $\mathbf{u}, \mathbf{v}, \mathbf{w}$ are unit vectors in hydrology modeling, find the maximum value of $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})$.

📌 Solution: The scalar triple product $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})$ represents the volume of the parallelepiped formed by the vectors. Its maximum absolute value occurs when the vectors are mutually orthogonal. Since they are unit vectors, the maximum value is $1 \cdot 1 \cdot 1 = 1$. For example, if $\mathbf{u} = \mathbf{v} = \mathbf{w}$ are orthonormal, the

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