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Vectors are essential elements in physics and many branches of mathematics. They provide a way to represent quantities that have both magnitude and direction. A vector is typically written as an ordered set of numbers, representing its components in space.
For example, in a three-dimensional space, a vector can be expressed with three components, such as \( (-1, 1, 2) \). Each component indicates the magnitude along a corresponding axis, commonly the x, y, and z axes.
Vectors are different from scalars, which have only magnitude. Understanding vectors is critical for problems involving direction and positioning, such as force, velocity, or displacement in space.
Knowing how to work with vectors sets the groundwork for many operations, including addition, subtraction, and, as seen in this exercise, the cross product.

Each vector is made up of components. These components are the individual parts that define the vector's direction and magnitude in a given space.
For vector \( A = (-1, 1, 2) \), the components are:

• \( a_1 = -1 \) along the x-axis
• \( a_2 = 1 \) along the y-axis
• \( a_3 = 2 \) along the z-axis

Likewise, vector \( B = (1, 0, -1) \) has components:

• \( b_1 = 1 \) along the x-axis
• \( b_2 = 0 \) along the y-axis
• \( b_3 = -1 \) along the z-axis

Understanding these components is vital when performing calculations such as the cross product. Each component plays a role in determining the resulting vector's position and direction.

Arithmetic operations with vectors involve combining their components in various ways to achieve a desired result. In this exercise, we focus on the cross product, an essential vector operation.
The cross product \( A \times B \) of two vectors produces another vector that is perpendicular to the plane formed by the original vectors.
To compute the cross product of vectors \( A \) and \( B \), we use the formula: \[ A \times B = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1) \] By substituting the components, we calculate:

• The first component: \( a_2b_3 - a_3b_2 = 1 \times (-1) - 2 \times 0 = -1 \)
• The second component: \( a_3b_1 - a_1b_3 = 2 \times 1 - (-1) \times (-1) = 2 - 1 = 1 \)
• The third component: \( a_1b_2 - a_2b_1 = (-1) \times 0 - 1 \times 1 = -1 \)

The resulting vector is \((-1, 1, -1)\). This process requires careful arithmetic to ensure that each component is calculated accurately. Understanding the operations involved in the cross product is crucial for solving problems involving vectors in physics and engineering.