Math Problem Statement

is it true that (u+v)x(u-v)=2(vxu)

Solution

The identity (u+v)×(uv)=2(v×u)(u+v) \times (u-v) = 2(v \times u) is not correct as stated. Here's the detailed reasoning:

Correct Cross Product Formula:

The cross product has the following distributive property:

(u+v)×(uv)=u×(uv)+v×(uv)(u+v) \times (u-v) = u \times (u-v) + v \times (u-v)

Expanding both terms, we get:

=u×uu×v+v×uv×v= u \times u - u \times v + v \times u - v \times v

Now, simplify:

  • u×u=0u \times u = 0 (since the cross product of a vector with itself is zero),
  • v×v=0v \times v = 0 (for the same reason),
  • u×v=(v×u)u \times v = -(v \times u) (since the cross product is anti-commutative).

Thus, the expression becomes:

0u×v+v×u00 - u \times v + v \times u - 0

Now, combine terms:

=(u×v)+(v×u)= -(u \times v) + (v \times u)

But, since (u×v)=v×u-(u \times v) = v \times u, we get:

(u+v)×(uv)=2(v×u)(u+v) \times (u-v) = 2(v \times u)

So, the correct identity is:

(u+v)×(uv)=2(v×u)(u+v) \times (u-v) = 2(v \times u)

Conclusion:

Yes, the equation is true: (u+v)×(uv)=2(v×u)(u+v) \times (u-v) = 2(v \times u).

Would you like more details or have any questions?


Here are 5 related questions for you to explore:

  1. What is the distributive property of the cross product?
  2. How does the cross product differ from the dot product in vector algebra?
  3. Can you prove that u×u=0u \times u = 0 for any vector uu?
  4. What is the geometric interpretation of the cross product?
  5. How does the cross product change if uu and vv 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