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Simple question: is the cross product defined in R^n ? In my linear algebra textbook, they talk about the dot product in length but don't even mention the cross product.
That's exactly what I was trying. And I think I'm on the right track to proving distributivity.Galileo said:That's a fairly difficult problem in general. Usually they start with the geometric definition, then show that the cross-product is distributive: A X (B+C)=(A X B)+(A X C)
Then you can derive the algebraic definition by writing the vectors out in components and use distributivity. Proving distributivity is not very easy, but certainly doable.
Good luckquasar987 said:That's exactly what I was trying. And I think I'm on the right track to proving distributivity.
The cross product in R^n is a vector operation that takes two vectors as inputs and produces a new vector that is perpendicular to both of the original vectors. It is denoted as a x b and can only be performed on vectors in three-dimensional space (R^3).
The cross product is defined as the determinant of a 3x3 matrix, where the first row consists of the unit vectors i, j, and k, the second row consists of the components of the first vector, and the third row consists of the components of the second vector. The resulting vector is the cross product of the two original vectors.
No, the cross product is only defined for vectors in three-dimensional space (R^3). This is because the determinant of a 3x3 matrix can only be calculated in R^3.
The cross product has a geometric interpretation as the area of the parallelogram formed by the two original vectors. The direction of the resulting vector is perpendicular to the plane formed by the two original vectors, according to the right-hand rule.
The cross product is undefined when the two original vectors are parallel or antiparallel. This is because the area of the parallelogram formed by parallel or antiparallel vectors is equal to 0, and therefore the cross product cannot be calculated.