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Find u · (v x w). This quantity is called the triple scalar product of u, v, and w. u = k, v = j, w = 2i

Find u · (v x w). This quantity is called the triple scalar product of u, v, and w. u = k, v = j, w = 2i

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Triple Scalar Product** To solve for \( \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) \), known as the triple scalar product of \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\), we use the following vector assignments: - \(\mathbf{u} = \mathbf{k}\) - \(\mathbf{v} = \mathbf{j}\) - \(\mathbf{w} = 2\mathbf{i}\) **Step-by-step Explanation:** 1. **Vector Cross Product:** - Calculate \(\mathbf{v} \times \mathbf{w}\). - Since \(\mathbf{v} = \mathbf{j}\) and \(\mathbf{w} = 2\mathbf{i}\), the cross product \(\mathbf{j} \times (2\mathbf{i})\) results in the vector \(-2\mathbf{k}\). 2. **Dot Product:** - Compute \(\mathbf{u} \cdot (-2\mathbf{k})\). - Since \(\mathbf{u} = \mathbf{k}\), the dot product \(\mathbf{k} \cdot (-2\mathbf{k})\) equals \(-2\). Therefore, the triple scalar product is \(-2\).
Transcribed Image Text:**Triple Scalar Product** To solve for \( \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) \), known as the triple scalar product of \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\), we use the following vector assignments: - \(\mathbf{u} = \mathbf{k}\) - \(\mathbf{v} = \mathbf{j}\) - \(\mathbf{w} = 2\mathbf{i}\) **Step-by-step Explanation:** 1. **Vector Cross Product:** - Calculate \(\mathbf{v} \times \mathbf{w}\). - Since \(\mathbf{v} = \mathbf{j}\) and \(\mathbf{w} = 2\mathbf{i}\), the cross product \(\mathbf{j} \times (2\mathbf{i})\) results in the vector \(-2\mathbf{k}\). 2. **Dot Product:** - Compute \(\mathbf{u} \cdot (-2\mathbf{k})\). - Since \(\mathbf{u} = \mathbf{k}\), the dot product \(\mathbf{k} \cdot (-2\mathbf{k})\) equals \(-2\). Therefore, the triple scalar product is \(-2\).
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