Math Problem Statement
If u × v = u × w and u ̸= 0, then does v = w? If so, justify your answer algebraically. If not, give a specific example of three nonzero vectors u, v, and w such that u × v = u × w but v ̸= w
Solution
To determine whether implies , let's break this problem down algebraically and geometrically.
Cross Product Algebraic Properties:
- The cross product of two vectors and results in a vector that is perpendicular to both and .
- If , we are given that the cross products of with and with are the same. This suggests that both and lie in a particular plane, but this doesn't necessarily mean that .
Step-by-step reasoning:
Given the equation , we can subtract both sides to get: This simplifies to: This equation means that the vector is parallel to the vector , or in other words, is parallel to . This does not imply that , but rather that: for some scalar .
Thus, the most general conclusion we can make is: This shows that and are not necessarily equal, but they differ by a scalar multiple of the vector .
Geometrical interpretation:
The vectors and can be different, but they must lie in the same plane that contains . So, could be shifted by some multiple of , while still producing the same cross product with .
Counterexample:
Here’s a specific example to show that is possible, even if .
Let: Now calculate the cross products:
So, , but clearly is not equal to .
Conclusion:
No, does not necessarily equal . The equation only implies that and differ by some multiple of , i.e., , but they are not necessarily the same vector.
Do you want further details, or have any questions? Here are five related questions to explore:
- How does the geometric interpretation of the cross product help understand vector equations?
- Can you provide another example where but ?
- What does imply about the relationship between vectors and ?
- How would this analysis change if we considered the dot product instead of the cross product?
- In which special cases would hold if ?
Tip: The cross product between two vectors results in a vector perpendicular to both. This makes it a valuable tool for determining orthogonality or relationships between planes and vectors.
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Math Problem Analysis
Mathematical Concepts
Vector Algebra
Cross Product
Linear Dependence
Vector Spaces
Formulas
u × v = u × w
u × (v - w) = 0
v = w + λu
Theorems
Cross Product Perpendicularity Theorem
Linear Dependence of Vectors
Suitable Grade Level
University/Advanced High School