91Ó°ÊÓ

The cross product is a mathematical operation used in vector mathematics to find a vector that is perpendicular to two given vectors. In our exercise, we are given the vectors \( \mathbf{u} = \mathbf{i} - \mathbf{k} \) and \( \mathbf{v} = \mathbf{j} + \mathbf{k} \). The goal is to find \( \mathbf{u} \times \mathbf{v} \). The cross product is represented by the symbol \( \times \) and involves using the determinant of a matrix that includes the unit vectors \( \mathbf{i}, \mathbf{j}, \) and \( \mathbf{k} \).

A convenient property of the cross product is that it provides a vector that is orthogonal to both original vectors. This is particularly useful in physics and engineering to find directions such as the orientation of a plane.

• It is defined only in three-dimensional space.
• The result of the cross product is another vector, not a scalar.
• Cross products can be calculated using a 3x3 determinant with the unit vectors and the vector components.

A 3D coordinate system is a three-dimensional space where any point can be specified by three coordinates (x, y, z). These coordinates correspond to the positions along the x-axis, y-axis, and z-axis respectively. The unit vectors \( \mathbf{i}, \mathbf{j}, \) and \( \mathbf{k} \) represent the positive directions of these axes.

In the exercise, the vector \( \mathbf{u} \) with components (1, 0, -1) is plotted from the origin, extending in the positive x-direction and negative z-direction. The vector \( \mathbf{v} \) starts at the same point, moving in both the positive y and z directions. Combined, their cross product, \( \mathbf{u} \times \mathbf{v} = \mathbf{i} - \mathbf{j} + \mathbf{k} \), is also plotted from the origin, showing us how the three vectors interact in three-dimensional space.

• 3D coordinate systems are essential for describing points and vectors in space.
• They help in visualizing and solving problems involving vectors.
• The axes form three perpendicular planes, making it simple to interpret vector directions.

Determinants play a crucial role in calculating the cross product of vectors. In our example, the cross product \( \mathbf{u} \times \mathbf{v} \) is calculated by evaluating the determinant of a 3x3 matrix:\[\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & 0 & -1 \ 0 & 1 & 1 \end{vmatrix}.\]

The determinant has a systematic way of breaking down these computations:

• Multiply the diagonal elements and subtract the product of the off-diagonal elements.
• This provides the vector components along the \( \mathbf{i}, \mathbf{j}, \) and \( \mathbf{k} \) directions.

The determinant is essential for understanding how vectors combine to form a cross product, reflecting differences in orientation and magnitude. It is a concise mathematical tool that simplifies complex vector operations.

Vector components are the building blocks of vectors, determining their magnitude and direction in each dimension. In our given example:

• The vector \( \mathbf{u} = \mathbf{i} - \mathbf{k} \) has components (1, 0, -1):
• 1 in the x-direction (represented by \( \mathbf{i} \))
• 0 in the y-direction (represented by \( \mathbf{j} \))
• -1 in the z-direction (represented by \( \mathbf{k} \))
• The vector \( \mathbf{v} = \mathbf{j} + \mathbf{k} \) has components (0, 1, 1):
• 0 in the x-direction
• 1 in the y-direction
• 1 in the z-direction

These components are used to determine how the vectors interact through operations such as the cross product. Understanding vector components is fundamental to recognizing how vectors behave in mathematical operations and in visualizing their interactions in space.