Is the Cross Product Cancellative?

Click For Summary

Discussion Overview

The discussion revolves around the question of whether the cross product is cancellative, specifically examining the implications of the equation u × v = u × w and whether it leads to the conclusion that v = w. The scope includes mathematical reasoning and exploration of vector properties.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant suggests that if u × v = u × w, then it follows that v = w.
  • Another participant counters this claim, arguing that the equality of the cross products does not imply the equality of the vectors due to potential differences in direction.
  • A third participant provides a counterexample using specific vector values to illustrate that the result of the cross product depends on certain components, indicating that x can vary without affecting the equality.
  • Further discussion points out that the equation u × v = u × w can be rewritten as u × (v - w) = 0, suggesting that u is parallel to the vector difference v - w.
  • Another participant raises the concern that if u is the zero vector, the original question becomes irrelevant, as the cross product would always yield zero.
  • A later reply reiterates the point about the zero vector, suggesting that the original question lacks significance in that context.

Areas of Agreement / Disagreement

Participants express disagreement regarding the implications of the cross product equality, with no consensus reached on whether the cross product is cancellative. The discussion remains unresolved with multiple competing views presented.

Contextual Notes

The discussion includes limitations related to the assumptions about the vectors involved, particularly the case when u is the zero vector, which complicates the analysis of the cross product's properties.

Jhenrique
Messages
676
Reaction score
4
If u × v = u × w, so v = w ?
 
Physics news on Phys.org
No.
At first, if two vectors are equal, then they are in the same direction so let's take their magnitudes. We have uv \sin\alpha=uw\sin\beta, so you have v\sin\alpha=w\sin\beta. Also \vec{v} and \vec{w} may differ in direction too.
 
  • Like
Likes   Reactions: 1 person
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.
 
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:
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?
 
D H said:
Not necessarily. What if u is the zero vector?

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

Similar threads

High School What is the cross product of two vectors?
  • · Replies 32 ·
2
Replies
32
Views
4K
Undergrad Equivalence of 2 definitions of the magnitude of the cross product?
  • · Replies 6 ·
Replies
6
Views
3K
Graduate Reason out the cross product (for the moment): a skew symmetric form
  • · Replies 14 ·
Replies
14
Views
4K
Undergrad Terminologies used to describe tensor product of vector spaces
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad The vector to which a dual vector corresponds
  • · Replies 13 ·
Replies
13
Views
4K
Undergrad Can the cross product concept be completely replaced by the exterior product?
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Dot product, inner product, and projections
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Question Regarding Definition of Tensor Algebra
  • · Replies 10 ·
Replies
10
Views
3K
Undergrad Question about the vector cross product in spherical or cylindrical coordinates
  • · Replies 5 ·
Replies
5
Views
5K
Undergrad How to approach a cross product question
  • · Replies 2 ·
Replies
2
Views
1K