91Ó°ÊÓ

The vector cross product is a fundamental operation in 3-dimensional vector algebra. It results in a vector that is perpendicular to the two original vectors. Let's say you have two vectors \(\mathbf{b}\) and \(\mathbf{c}\). The cross product of these vectors, denoted as \(\mathbf{b} \times \mathbf{c}\), follows the right-hand rule for determining the direction of the resulting vector.

In calculating the cross product, we use the determinant of a special 3x3 matrix. This matrix consists of the unit vectors \(\mathbf{i}, \mathbf{j},\) and \(\mathbf{k}\) in its first row, the components of vector \(\mathbf{b}\) in its second row, and the components of vector \(\mathbf{c}\) in its third row. It looks like this:

\[\mathbf{b} \times \mathbf{c} = \left| \begin{array}{ccc}\mathbf{i} & \mathbf{j} & \mathbf{k} \1 & 4 & 1 \1 & 1 & 5 \end{array} \right|\] The cross product is useful in physics and engineering, especially when dealing with rotational vectors.

When computed, the cross product \(\mathbf{b} \times \mathbf{c}\) gives you a new vector which is: \(19\mathbf{i} - 4\mathbf{j} - 3\mathbf{k}\). This new vector is essential in calculating other important quantities such as volume.

The scalar triple product is a very interesting operation involving three vectors. It is crucial for finding the volume of a parallelepiped formed by those vectors. For vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\), the scalar triple product is represented as \(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})\). This operation results in a scalar, which in the context of this problem, provides the volume of the parallelepiped.

A helpful aspect of the scalar triple product is its geometric interpretation. It essentially calculates how much vector \(\mathbf{a}\) "projects" onto the vector generated by \(\mathbf{b} \times \mathbf{c}\), giving a scalar measure related to the volume. The calculation in this exercise yielded a scalar value of 50, thus the volume of the parallelepiped is 50 units cubed.

This operation highlights an important relationship between vectors in three-dimensional space, and is not only important in geometry but also in physics, where similar calculations are used for flux, torque, and other applications.

Determinants play a crucial role in vector calculus, especially when dealing with linear transformations and systems of equations. In our problem, the determinant helps us find the cross product of vectors \(\mathbf{b}\) and \(\mathbf{c}\).

The determinant of a 3x3 matrix is the volume factor when transforming geometric objects. It's computed as the sum of the products of its diagonal elements minus the products of its counter-diagonal elements. In this exercise, employing the determinant for \(\mathbf{b} \times \mathbf{c}\) involved the unit vectors and the components of \(\mathbf{b}\) and \(\mathbf{c}\). You can see the formula applied as:

\[\begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \1 & 4 & 1 \1 & 1 & 5 \end{vmatrix}\]

This gave us the new vector \(19\mathbf{i} - 4\mathbf{j} - 3\mathbf{k}\). By following the components of the determinant effectively, you achieve a powerful tool capable of facilitating complex vector operations crucial in multi-dimensional analysis.