Math Problem Statement
Suppose z = u × v where × is the cross-product. Prove or provide a counter example of the fact that u, v and z are linearly independent.
Solution
Let us first recall some facts about the cross product and linear independence.
Definitions:
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Cross product: Given two vectors and in , their cross product is a vector perpendicular to both and .
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Linear independence: A set of vectors is linearly independent if the only solution to the equation is .
Approach to the problem:
We need to determine whether , , and are always linearly independent, or if there is a counterexample where they are linearly dependent.
Step 1: Special Case Analysis
Consider two simple cases that could break linear independence:
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Case 1: and are parallel. If and are parallel vectors, then their cross product is zero, i.e., . Hence, the set .
In this case, and are linearly dependent because for some scalar . Therefore, the set is linearly dependent.
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Case 2: and are non-parallel. If and are not parallel, then , and the resulting vector is perpendicular to both and . Since is perpendicular to the plane containing and , the vectors , , and should be linearly independent.
To prove this, assume there are scalars such that: Dot this equation with , and using the fact that , we get: Similarly, dotting the equation with gives: Solving these two equations will lead to and if and are not parallel. Finally, since , this also implies , proving that the vectors are linearly independent.
Conclusion:
- If and are parallel, , , and are linearly dependent because .
- If and are not parallel, , , and are linearly independent.
Thus, the set is not always linearly independent, and a counterexample occurs when and are parallel.
Would you like further details or explanations on any part of this?
Related Questions:
- What are some geometric interpretations of the cross product?
- How can you use the determinant to check for linear independence?
- How does the dot product relate to the concept of linear dependence?
- In what cases does the magnitude of the cross product equal zero?
- How would linear independence change in higher dimensions?
Tip:
If two vectors are parallel, their cross product will always be zero, indicating that they are linearly dependent.
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Vector Calculus
Formulas
Cross product formula: u × v = z
Linear independence equation: c1u + c2v + c3z = 0
Theorems
Linear Independence Theorem
Cross Product Properties
Suitable Grade Level
College level (Undergraduate)