# Cross/Dot Product in n dimensions

• rachbomb
In summary: However, proving the equation in R^7 is a bit more involved than just applying the usual rules of algebra.
rachbomb
I would like to show that (cxb).a = (axc).b in Rn where x denotes the cross product and . denotes the dot product.

Since the cross product is only defined in R1, R3, and R7, my inclination is to prove the above equation in cases (case one being a,b,c are vectors in R1, etc). However, this seems a bit tedious, particularly for R7.

I am familiar with the Triple Product in R3, but am unsure if it applies in Rn, and if so how to prove it. If so, this seems like a much quicker and more concise proof.

rachbomb said:
I would like to show that (cxb).a = (axc).b in Rn where x denotes the cross product and . denotes the dot product.

Since the cross product is only defined in R1, R3, and R7,
I've never heard of the cross product being defined in any space other than R3.
rachbomb said:
my inclination is to prove the above equation in cases (case one being a,b,c are vectors in R1, etc). However, this seems a bit tedious, particularly for R7.

I am familiar with the Triple Product in R3, but am unsure if it applies in Rn, and if so how to prove it. If so, this seems like a much quicker and more concise proof.

You might want to look into the wedge product (aka exterior algebra) as the generalization of the cross product to $$R^n$$. However, the result of the product is a tensor, not a vector. By virtue of the property $$u \wedge v = - v \wedge u$$, it satisfies $$u \wedge v \wedge w = w \wedge u \wedge v$$. In 3d, you can use the Hodge dual to identify the vector $$*(u\wedge v)$$ with the cross product and $$*(u \wedge v \wedge w )$$ with $$(u\times v)\cdot w$$.

To Mark44, it's possible to define a cross product in $$R^7$$ because of the existence of the octonions. There's a construction of that product that's analogous to the relationship between the cross product in 3d and the quaternion algebra.

## 1. What is the difference between cross product and dot product?

The cross product is a vector operation that results in a vector that is perpendicular to both of the input vectors. It is denoted by the symbol "×" and is defined as A × B = |A| |B| sin(θ) n, where θ is the angle between the two vectors and n is a unit vector perpendicular to the plane containing A and B. On the other hand, the dot product is a scalar operation that results in a single number. It is denoted by the symbol "⋅" and is defined as A ⋅ B = |A| |B| cos(θ), where θ is the angle between the two vectors. In summary, the cross product produces a vector while the dot product produces a scalar.

## 2. How is the cross product calculated in n dimensions?

The cross product can be calculated in n dimensions using the vector cross product formula for 3 dimensions. This formula can be extended to n dimensions by using the determinant of an (n+1) by (n+1) matrix. The resulting vector will have n components, with the first (n-1) components being zero. This means that the cross product in n dimensions is only defined for n-1 dimensional space.

## 3. What is the geometric interpretation of the cross product in n dimensions?

In 3 dimensions, the cross product can be interpreted as the area of the parallelogram formed by the two input vectors. In n dimensions, the cross product can be interpreted as the (n-1) dimensional volume of the parallelepiped formed by the n-1 input vectors. This means that the cross product in n dimensions is also known as the wedge product.

## 4. How is the dot product used in vector calculus?

In vector calculus, the dot product is used to calculate the gradient of a scalar field. It is also used in the definition of the divergence and curl of a vector field. In addition, the dot product is used in the equations of motion for particles and rigid bodies in physics.

## 5. Can the cross product and dot product be used in any number of dimensions?

The cross product can only be used in 3 dimensions or in spaces with an odd number of dimensions. This is because the cross product only exists for n-1 dimensional space. On the other hand, the dot product can be used in any number of dimensions as it only involves multiplying the magnitudes of the two input vectors and the cosine of the angle between them.

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