Determining the Dot Product from Cross Product: An In-Depth Analysis

When working with vectors in linear algebra, the dot product and cross product are two essential operations. This article explores the relationship between these operations and investigates whether we can determine the dot product given the cross product. Specifically, we will address the scenario where 'cross' denotes the outer or tensor product of two real vectors of the same dimension.

Overview of Vector Products

In linear algebra, there are three primary definitions of the product of two vectors: the cross product, the dot product, and the outer or tensor product. The dot product yields a scalar value, whereas the cross and outer products yield vectors.

Dot Product: A Scalar Value

The dot product of two vectors, denoted as a and b, which belong to R^n, is a scalar value defined as:

a cdot b sum_{k1}^n a_k cdot b_k

This can be expressed using matrix notation, where a^T represents the transpose of vector a:

a cdot b a^T cdot b

Example Calculation

Let's consider two vectors a [-3, -1, 5] and b [-2, 1, 4]. The dot product of these vectors is:

a cdot b (-3 cdot -2) (-1 cdot 1) (5 cdot 4) 6 - 1 20 25

Outer or Tensor Product

The outer or tensor product of two vectors b and a is a matrix formed by multiplying each element of b with each element of a. This product is denoted as:

b otimes a b cdot a^T

For example, if b [-1, 5] and a [2, -2], the outer product is:

b otimes a begin{bmatrix} -1 cdot 2 5 cdot 2 -1 cdot -2 5 cdot -2 end{bmatrix} begin{bmatrix} -2 10 2 -10 end{bmatrix}

Relationship Between Dot and Outer Products

In the context of R^n, the outer product of vectors a and b yields a square matrix. The dot product can be recovered from the trace of this matrix, which is the sum of its diagonal elements:

a cdot b text{tr}(a otimes b) sum_{k1}^n (a otimes b)_{kk}

Summary and Limitations

The dot product can indeed be determined from the outer product when the vectors are of the same dimension. However, this relationship does not work in reverse. Knowing the cross product alone does not allow us to determine the dot product because we need the norms of the vectors, which are not directly provided by the cross product.

Conclusion

While the relationship between the dot and outer products is well-established in linear algebra, using the cross product to find the dot product is more complex. Understanding these operations is crucial for advanced applications in fields such as computer graphics, physics, and engineering.