Is the Cross Product Cancellative?

In summary, the conversation discusses the relationship between two vectors, u and v, and whether or not they are equal based on their magnitudes and direction. The conclusion is that if u × v = u × w, it does not necessarily mean that v = w, as it is possible for u to be the zero vector.
  • #1
Jhenrique
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If u × v = u × w, so v = w ?
 
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  • #2
No.
At first, if two vectors are equal, then they are in the same direction so let's take their magnitudes. We have [itex]uv \sin\alpha=uw\sin\beta [/itex], so you have [itex] v\sin\alpha=w\sin\beta [/itex]. Also [itex] \vec{v} [/itex] and [itex] \vec{w} [/itex] may differ in direction too.
 
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  • #3
You should have been able to find a simple counterexample. E.g. if u = (1, 0, 0):

(1, 0, 0) x (x, y, z) = (0, -z, y)

So, the result only depends on y and z, and x can be anything.
 
  • #4
Jhenrique said:
If u × v = u × w, so v = w ?

u × v = u × w is equivalent to u × (v-w) =0.
Therefore u is parallel to v-w, so that v-w is a multiple of u.
 
Last edited:
  • #5
mathman said:
u × v = u × w is equivalent to u × (v-w) =0.
Therefore u is parallel to v-w, so that v-w is a multiple of u.
Not necessarily. What if u is the zero vector?
 
  • #6
D H said:
Not necessarily. What if u is the zero vector?

Quibble. The original question is pointless for u=0.
 

FAQ: Is the Cross Product Cancellative?

What is the definition of "Cross product is cancellative"?

The cross product is a mathematical operation that takes two vectors as input and produces a third vector that is perpendicular to both input vectors. When we say that the cross product is cancellative, we mean that if the cross product of two vectors is equal to the cross product of two other vectors, then the original two vectors must also be equal.

Why is "Cross product is cancellative" important in mathematics?

The cancellative property of the cross product allows us to simplify equations involving vectors and makes it easier to solve problems in physics and engineering. It also helps us to identify equivalent vectors and simplify geometric proofs.

Are there any exceptions to the "Cross product is cancellative" property?

Yes, there are some exceptions to this property. It only holds true for three-dimensional vectors and does not apply to vectors in higher dimensions. Additionally, the property may not hold if the vectors are parallel or collinear.

How can we prove that the "Cross product is cancellative" property holds true?

The cancellative property can be proven using the properties of determinants and the properties of vector multiplication. By expanding the determinants of the cross products, we can show that they are equal only if the original vectors are also equal.

What are some real-world applications of the "Cross product is cancellative" property?

The cross product is cancellative property has various real-world applications in physics, engineering, and computer graphics. It is used in calculating the torque exerted by a force on a rigid body, determining the direction of magnetic fields, and in 3D computer graphics to calculate lighting and shading effects on objects.

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