Question

Compute the product ab by the definition of the product of matrices. Compute a and b separately, and then use the row-column rule for computing ab.

Answer 1

The product of matrices A and B can refer to either the scalar product or the cross product; the latter is computed using the formula with components and unit vectors, while maintaining the specific multiplication order due to the anticommutative property.

Step-by-step explanation:

Computing the Product of Matrices ab

In mathematics, specifically in vector calculus, the product of matrices is a fundamental operation. When working with vectors, there are two types of products that can be computed: the scalar (or dot) product and the cross product. The scalar product is computed as A.B = Ax Bx + Ay By + Az Bz, where A and B are vectors and the subscript x, y, z denotes their respective components along the coordinate axes.

The cross product A x B results in a vector that is perpendicular to the plane containing vectors A and B, and is computed using the following definition:

Č = Ả x B = (Ay Bz – Az By)Î + (Az Bx – Ax Bz)Ą + (Ax By – AyBx)Ê.

To compute the cross product, it's important to note the anticommutative property, meaning that A x B is not the same as B x A. Instead, B x A yields a vector that points in the exact opposite direction to A x B. Additionally, in practice, various equations can simplify the computation, but care should be taken to ensure that the correct order of multiplication is preserved to get the correct direction of the resulting vector.

Answer 2

To compute the matrix product ab, one must multiply each entry from the rows of the first matrix by the corresponding entries in the columns of the second matrix and then add those products. The cross product formula is relevant here, and order matters due to the anticommutative property. Scalar products are simple summations of scalar multiples and are used for finding angles between vectors.

Step-by-step explanation:

Computing the Product of Matrices

To compute the product of matrices ab by definition, we first need to understand that the product of two matrices involves a series of multiplications and additions of their entries, commonly referred to as the dot product when considering vectors. This process is represented by multiplying each entry from the rows of the first matrix by the corresponding entries in the columns of the second matrix and summing those products. It is essential when performing these operations, especially in the context of the cross product, to adhere to the specific properties including anticommutative property, which means that the order of multiplication matters, and the result will change if the order is reversed.

Scalar product (dot product) is a fundamental concept in computing matrix products, where only the corresponding components are multiplied and then added together. However, calculating the cross product requires crossing two vectors to result in a third vector that is perpendicular to the plane formed by the first two vectors. The formula for the cross product is as follows:
Č = Ả × B = (Ay B₂ – Az By)Î + (Az Bx − Ax Bz)Ĵ + (Ax By –− AyBx)Ê.

When using unit vectors, the scalar product simplifies because the product of two different unit vectors is zero, and the product of a unit vector with itself is one. Therefore, the scalar product can be expressed as Ả B = Ax Bx + Ay By + A, B,. This simple addition of scalar multiples is crucial when we need to find an angle between two vectors using scalar product.