Math Problem Statement
is it true that (u+v)x(u-v)=2(vxu)
Solution
The identity is not correct as stated. Here's the detailed reasoning:
Correct Cross Product Formula:
The cross product has the following distributive property:
Expanding both terms, we get:
Now, simplify:
- (since the cross product of a vector with itself is zero),
- (for the same reason),
- (since the cross product is anti-commutative).
Thus, the expression becomes:
Now, combine terms:
But, since , we get:
So, the correct identity is:
Conclusion:
Yes, the equation is true: .
Would you like more details or have any questions?
Here are 5 related questions for you to explore:
- What is the distributive property of the cross product?
- How does the cross product differ from the dot product in vector algebra?
- Can you prove that for any vector ?
- What is the geometric interpretation of the cross product?
- How does the cross product change if and are parallel vectors?
Tip: The cross product is always perpendicular to the two vectors involved, and its magnitude depends on the sine of the angle between them.
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Math Problem Analysis
Mathematical Concepts
Vector Algebra
Cross Product
Distributive Property
Formulas
(u+v) × (u-v) = u × (u-v) + v × (u-v)
u × v = -(v × u)
Theorems
Anti-commutative property of the cross product
Cross product of a vector with itself is zero
Suitable Grade Level
Undergraduate level
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