Question

If \(\mathbf{i}, \mathbf{j},\) and \(\mathbf{k}\) are the coordinate vectors, verify that \(\mathbf{i} \times \mathbf{j}=\mathbf{k}, \mathbf{j} \times \mathbf{k}=\mathbf{i},\) and \(\mathbf{k} \times \mathbf{i}=\mathbf{j}\)

Short answer

The cross products verify the given equations are true: \(\mathbf{i} \times \mathbf{j} = \mathbf{k}\), \(\mathbf{j} \times \mathbf{k} = \mathbf{i}\), and \(\mathbf{k} \times \mathbf{i} = \mathbf{j}\).

Step-by-step solution

Understanding the Coordinate Vectors

The coordinate vectors are unit vectors that correspond to the x, y, and z axes in a 3D Cartesian coordinate system. Specifically, \(\mathbf{i} = \langle 1, 0, 0 \rangle\), \(\mathbf{j} = \langle 0, 1, 0 \rangle\), and \(\mathbf{k} = \langle 0, 0, 1 \rangle\).

Cross Product Definition

The cross product of two vectors \(\mathbf{a} = \langle a_1, a_2, a_3 \rangle\) and \(\mathbf{b} = \langle b_1, b_2, b_3 \rangle\) is given by \(\mathbf{a} \times \mathbf{b} = \langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle\).

Calculate \(\mathbf{i} \times \mathbf{j}\)

Using the cross product formula, find \(\mathbf{i} \times \mathbf{j}\):\[ \mathbf{i} \times \mathbf{j} = \langle 1, 0, 0 \rangle \times \langle 0, 1, 0 \rangle \]Thus, the components are:- \(0 \times 0 - 0 \times 1 = 0\)- \(0 \times 1 - 1 \times 0 = 0\)- \(1 \times 1 - 0 \times 0 = 1\)Therefore, \(\mathbf{i} \times \mathbf{j} = \langle 0, 0, 1 \rangle = \mathbf{k}\).

Calculate \(\mathbf{j} \times \mathbf{k}\)

Using the cross product formula, find \(\mathbf{j} \times \mathbf{k}\):\[ \mathbf{j} \times \mathbf{k} = \langle 0, 1, 0 \rangle \times \langle 0, 0, 1 \rangle \]Thus, the components are:- \(1 \times 1 - 0 \times 0 = 1\)- \(0 \times 0 - 0 \times 1 = 0\)- \(0 \times 0 - 1 \times 0 = 0\)Therefore, \(\mathbf{j} \times \mathbf{k} = \langle 1, 0, 0 \rangle = \mathbf{i}\).

Calculate \(\mathbf{k} \times \mathbf{i}\)

Using the cross product formula, find \(\mathbf{k} \times \mathbf{i}\):\[ \mathbf{k} \times \mathbf{i} = \langle 0, 0, 1 \rangle \times \langle 1, 0, 0 \rangle \]Thus, the components are:- \(0 \times 0 - 1 \times 0 = 0\)- \(1 \times 1 - 0 \times 0 = 1\)- \(0 \times 0 - 0 \times 1 = 0\)Therefore, \(\mathbf{k} \times \mathbf{i} = \langle 0, 1, 0 \rangle = \mathbf{j}\).

Verification Summary

We have verified the relationships by calculating the cross products and found that: - \(\mathbf{i} \times \mathbf{j} = \mathbf{k}\)- \(\mathbf{j} \times \mathbf{k} = \mathbf{i}\)- \(\mathbf{k} \times \mathbf{i} = \mathbf{j}\)This confirms the given statements are true.

Key concepts

coordinate vectors

Coordinate vectors are vital components in mathematics and physics, particularly when dealing with vectors in dimensional spaces. In a 3D Cartesian coordinate system, you will encounter the standard coordinate vectors, which are also known as the unit vectors. These vectors are:

• \( \mathbf{i} = \langle 1, 0, 0 \rangle \)
• \( \mathbf{j} = \langle 0, 1, 0 \rangle \)
• \( \mathbf{k} = \langle 0, 0, 1 \rangle \)

These vectors act as the building blocks for any vector in 3D space. By combining them with scalar values, you can represent any vector within this space.
For instance, any vector \( \mathbf{v} \) can be expressed as \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k} \), where \( a, b, \) and \( c \) are scalar quantities.
Understanding these coordinate vectors is essential because they help you simplify complex vector calculations and visualizations.

3D Cartesian coordinate system

A 3D Cartesian coordinate system is a method for graphically representing points or vectors in three-dimensional space. This system leverages three axes, which are mutually perpendicular to each other: the x-axis, y-axis, and z-axis.
Each point in this space is described by a triplet coordinate \((x, y, z)\), with each number representing the location of the point along one of the axes.

• The x-axis typically represents the horizontal dimension.
• The y-axis represents the vertical dimension.
• The z-axis represents the depth or forward-backward dimension.

The 3D Cartesian coordinate system is fundamental in disciplines such as engineering, physics, and computer graphics because it provides a simple yet effective way to model real-world objects and interactions in a spatial environment.
In this system, vector operations like the cross product rely on these axes, defining interactions between vectors, such as rotation and directionality.

unit vectors

Unit vectors are vectors with a magnitude of 1. They are primarily used to indicate direction without considering magnitude. In the context of a 3D Cartesian coordinate system, the unit vectors \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) represent movement solely along the x, y, and z axes, respectively.
The length or magnitude of each of these vectors is exactly 1, which means they do not stretch or shrink the vector they describe.

• \( \| \mathbf{i} \| = 1 \)
• \( \| \mathbf{j} \| = 1 \)
• \( \| \mathbf{k} \| = 1 \)

Often, they serve as a reference point for deciding directions in space, and any vector in space can be broken down into components along \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \).
Beyond these standard unit vectors in a 3D space, unit vectors can be created from any vector by dividing the vector by its magnitude. This transformation results in a new vector that points in the same direction but has a magnitude of one.