- #1

Kevin McHugh

- 318

- 164

i.e. e

_{1}*e

_{1}= 1

e

_{1}*e

_{2}= 0

e

_{1}x e

_{2}= e

_{3}

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- I
- Thread starter Kevin McHugh
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In summary: So we can call the cross product a "vector operation", but we can't do that with the wedge product.In summary, the wedge product is a bilinear operation that is associative and follows certain rules, such as the property that the wedge product of a vector with itself is zero. It is similar to the cross product, but produces a different mathematical structure and cannot be considered a vector operation.

- #1

Kevin McHugh

- 318

- 164

i.e. e

e

e

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- #2

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\underbrace{V \wedge V \wedge \ldots \wedge V}_{n\ times}## and ##b \in \Lambda^m(V)= \underbrace{V \wedge V \wedge \ldots \wedge V}_{m\ times}##

- #3

Kevin McHugh

- 318

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Are those components or basis vectors? Can you explain that in English?

- #4

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If we have a vector space ##V## over a field ##\mathbb{F}## in which ##1+1 \neq 0## holds, then for ## a,b,c \in V## and ##\lambda \in \mathbb{F}## the following is true:

- ##(a+b) \wedge c = a\wedge c + b \wedge c##
- ##\lambda (a \wedge b) = (\lambda a) \wedge b = a \wedge (\lambda b)##
- ##a \wedge (b \wedge c) = (a \wedge b) \wedge c##
- ##a \wedge a = 0##
- ##a_1 \wedge \ldots \wedge a_n \wedge b_1 \wedge \ldots \wedge b_m = (-1)^{nm} b_1 \wedge \ldots \wedge b_m \wedge a_1 \wedge \ldots \wedge a_n##

Perhaps you want to read the Wikipedia entry on it: https://en.wikipedia.org/wiki/Exterior_algebra

- #5

Stephen Tashi

Science Advisor

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Kevin McHugh said:Are those components or basis vectors?

The first question is whether ##a \wedge b## is something (e.g. a vector or a scalar) in the same vector space that contains ##a## and ##b##.

I'd say no. For ##a## and ##b## in a a vector space ##V##, in order to define ##a \wedge b##, you must define a different mathematical structure than ##V## itself. ( As an analogy, we can use two real numbers x1, x2 to define an interval [x1,x2], but "an interval" is a different thing than a single real number. )

By contrast, the cross product operation (in 3 dimensions) ##a \times b##

The wedge product of basis vectors is a mathematical operation used in multilinear algebra where two or more vectors are combined to form a new vector. It is also known as the exterior product or outer product.

The wedge product and the dot product are two different operations in vector algebra. While the dot product results in a scalar value, the wedge product results in a vector. Additionally, the dot product is commutative, while the wedge product is anti-commutative.

The wedge product can be thought of as a measure of the area of a parallelogram formed by two vectors. The magnitude of the resulting vector is equal to the area of the parallelogram, and the direction of the vector is perpendicular to the plane formed by the two vectors.

In physics, the wedge product is used to calculate the work done by a force on an object. It is also used in electromagnetism to calculate the flux of a vector field through a surface.

Yes, the wedge product can be extended to any number of vectors. The resulting vector will be perpendicular to the hyperplane formed by all the input vectors, and its magnitude will be equal to the volume of the parallelepiped formed by the vectors.

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