91Ó°ÊÓ

The cross product of two vectors is a fundamental operation in vector calculus, often denoted as \( \mathbf{a} \times \mathbf{b} \). It results in a vector that is perpendicular to the plane containing the original vectors. This makes cross products particularly useful in physics and engineering, where understanding orientations and angles in three dimensions is crucial.

To compute the cross product, you'll often use the determinant of a specific matrix set up. The result of the cross product not only gives us a vector that is orthogonal to the input vectors but also indicates the area of the parallelogram spanned by the vectors. It's important to note that the cross product is defined only in three-dimensional space. In our exercise, the cross product \( \mathbf{a} \times \mathbf{b} \) results in the vector \(-\mathbf{i} + \mathbf{j} + 0\mathbf{k}\). This indicates that the resultant vector is orthogonal to both \( \mathbf{a} \) and \( \mathbf{b} \).

In the context of our exercise, to find the cross product, we need to calculate the determinant of a 3x3 matrix. The determinant is a special number that can be calculated from the elements of a square matrix. It provides important information about the matrix properties, like whether the matrix is invertible and the volume scaling factor.

For any 3x3 matrix, the determinant can be calculated using the rule of Sarrus or cofactor expansion. In our case, the matrix:

• \( \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & 1 & 3 \ 1 & 1 & 2 \end{vmatrix} \)

gives us the structure needed to use cofactor expansion, where the determinant helps find the cross product. Understanding how to compute determinants is essential because they are a stepping stone in many areas of advanced mathematics and applied sciences.

Cofactor expansion (or Laplace expansion) is a mathematical technique used for calculating determinants of matrices, which is critical in vectors calculus when finding the cross product. To use cofactor expansion to compute the determinant of our 3x3 matrix, we expand along the top row where the standard unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) are located.

This involves:

• For each entry in the row, removing the row and column containing that entry and calculating the determinant of the remaining 2x2 matrix.
• Multiplying these 2x2 determinants by their respective cofactor, which includes a sign based on its position \((-1)^{i+j}\).

In our solution, the cofactor expansion was used, resulting in the calculated determinants \(: \mathbf{i}(1 \cdot 2 - 1 \cdot 3) - \mathbf{j}(1 \cdot 2 - 1 \cdot 3) + \mathbf{k}(1 \cdot 1 - 1 \cdot 1)\), which then simplifies to \(-\mathbf{i} + \mathbf{j} + 0\mathbf{k}\). This process clearly demonstrates the application of cofactor expansion in vector calculus.

Matrices play a critical role in vector calculus, especially when it comes to operations involving multiple dimensions. In our exercise, we constructed a 3x3 matrix with the basis vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) and the components of the vectors \( \mathbf{a} \) and \( \mathbf{b} \). This matrix is crucial for computing the cross product, as it elegantly aligns the elements necessary for the determinant calculation.

A matrix is essentially a rectangular arrangement of numbers in rows and columns, capable of representing linear transformations and systems of linear equations. It is a powerful framework because it can contain the information about entire spaces, transformations, or operations within its structure.

In this exercise, the matrix set up led directly to solving the cross product using determinant and cofactor expansion methods, illustrating the deep interconnectedness of these mathematical concepts. Understanding matrices is vital not just for vector calculus, but also for further exploration in subjects such as linear algebra and differential equations.