Transcribed Image Text:**Cross Product and Orthogonality**
**Problem:**
Find **u × v.**
Given:
- \(\mathbf{u} = \mathbf{j} + 4\mathbf{k}\)
- \(\mathbf{v} = 2\mathbf{i} - \mathbf{k}\)
[Box for solution]
**Show that \(\mathbf{u} \times \mathbf{v}\) is orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\).**
Calculate:
- \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{u} =\) [Box for solution]
- \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{v} =\) [Box for solution]
**Explanation:**
- **u × v** is the cross product of vectors **u** and **v**, which results in a vector perpendicular to both.
- Orthogonality is verified by showing that the dot product of **u × v** with both **u** and **v** equals zero.
Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.
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![**Cross Product and Orthogonality**
**Problem:**
Find **u × v.**
Given:
- \(\mathbf{u} = \mathbf{j} + 4\mathbf{k}\)
- \(\mathbf{v} = 2\mathbf{i} - \mathbf{k}\)
[Box for solution]
**Show that \(\mathbf{u} \times \mathbf{v}\) is orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\).**
Calculate:
- \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{u} =\) [Box for solution]
- \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{v} =\) [Box for solution]
**Explanation:**
- **u × v** is the cross product of vectors **u** and **v**, which results in a vector perpendicular to both.
- Orthogonality is verified by showing that the dot product of **u × v** with both **u** and **v** equals zero.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a7c37e1-d117-46e5-8cc2-cb36983cacd6%2F60c4b8ff-9945-4857-8398-fe60094310a6%2Fgfbenvo_processed.png&w=3840&q=75)
