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When working with vectors, the coordinate axes are essential references. In a standard three-dimensional Cartesian coordinate system, there are three axes:

• x-axis: This is usually the horizontal axis, pointing to the right.
• y-axis: The vertical axis, often pointing upwards.
• z-axis: This axis comes out of the plane formed by the x and y axes. It usually points towards us.

These axes intersect at a common point known as the origin, denoted as (0, 0, 0). The coordinate axes allow us to visualize and describe the positions and directions of vectors in space.
Each vector can be expressed in terms of these axes using its components. For example, a vector \( \mathbf{u} \) can be expressed as \( \mathbf{u} = u_x \mathbf{i} + u_y \mathbf{j} + u_z \mathbf{k} \), where \( \mathbf{i}, \mathbf{j}, \) and \( \mathbf{k} \) are unit vectors along the x, y, and z axes respectively.

Unit vectors are vectors with a magnitude of one. They are used to denote directions along the coordinate axes and are the building blocks for any vector in three-dimensional space. The three main unit vectors are:

• \( \mathbf{i} \): Represents a unit vector in the direction of the x-axis. Its coordinates are \((1, 0, 0)\).
• \( \mathbf{j} \): Represents a unit vector in the direction of the y-axis, having coordinates \((0, 1, 0)\).
• \( \mathbf{k} \): Represents a unit vector in the direction of the z-axis. Its coordinates are \((0, 0, 1)\).

These unit vectors give us a simple way to describe and draw vectors on the coordinate axes. For example, in the given exercise, \( \mathbf{u} = \mathbf{i} \) means it lies entirely along the x-axis, and \( \mathbf{v} = \mathbf{j} \) lies along the y-axis. Any arbitrary vector can be constructed by scaling and adding these unit vectors together.

The determinant plays a key role in finding the cross product of two vectors. A determinant is a special number calculated from a square matrix. For three-dimensional vectors, the cross product involves a 3x3 determinant:\[\det \begin{pmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ u_x & u_y & u_z \ v_x & v_y & v_z \end{pmatrix} \]This setup allows us to compute the vector \( \mathbf{u} \times \mathbf{v} \). The determinant essentially helps us derive the new vector, which is perpendicular to both original vectors. For our specific vectors \( \mathbf{u} = \mathbf{i} \) and \( \mathbf{v} = \mathbf{j} \), the determinant method yields the result \( \mathbf{k} \), showing that the cross product points in the positive z-direction. Understanding the mechanics of the determinant makes calculating cross products straightforward.

Three-dimensional vectors are essential components in physics, engineering, and mathematics. They have distinct properties that make them powerful tools for representing quantities in space. A three-dimensional vector can be written as \( \mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j} + v_z \mathbf{k} \), where \( v_x, v_y, \) and \( v_z \) are the components of the vector along the respective axes.

• Length: Calculated using the formula \( \sqrt{v_x^2 + v_y^2 + v_z^2} \) which gives the Euclidean distance from the origin.
• Direction: Indicated by the unit vector form of the vector, which is given by normalizing it i.e. dividing each component by the vector's length.

In our specific case, the vectors \( \mathbf{u} \) and \( \mathbf{v} \) are already unit vectors, meaning their length is 1. The cross product \( \mathbf{u} \times \mathbf{v} \) results in \( \mathbf{k} \), a vector that is perpendicular to the x and y axes, showing the intriguing nature of three-dimensional vector interactions.