91Ó°ÊÓ

Orthogonality is a mathematical condition where two vectors are perpendicular to each other. In this exercise, we are tasked with finding a homogeneous linear system where vectors in \(R^3\), denoted by \(\mathbf{x} = (x_1, x_2, x_3)\), must be orthogonal to the given vectors \(\mathbf{a} = (-3, 2, -1)\) and \(\mathbf{b} = (0, -2, -2)\). To determine orthogonality, we use the dot product rule: two vectors are orthogonal if their dot product is zero. This leads us to establish the two conditions:

• \(\mathbf{a} \cdot \mathbf{x} = -3x_1 + 2x_2 - x_3 = 0\)
• \(\mathbf{b} \cdot \mathbf{x} = -2x_2 - 2x_3 = 0\)

These equations express the need for the solution vector \(\mathbf{x}\) to be orthogonal to both \(\mathbf{a}\) and \(\mathbf{b}\), forming the basis of our next steps in defining the system.

The solution space of a homogeneous linear system refers to the set of all vectors that satisfy the given linear equations. After establishing the orthogonality conditions, we translate them into equations: 1. \(-3x_1 + 2x_2 - x_3 = 0\) and 2. \(x_2 + x_3 = 0\). These equations must be solved simultaneously to understand the vectors \(\mathbf{x} = (x_1, x_2, x_3)\) that satisfy both conditions. By substituting \(x_2 = -x_3\) from the second equation into the first, we simplify the system to a single parameter representation. This reduction helps identify the set of vectors forming the solution space. Solving these provides insight into how vectors can be manipulated in a space defined by linear equations and is crucial for deeper exploration.

Finding the general solution of a linear system involves expressing the solutions in terms of one or more parameters. Here, the reduced form \(-3x_1 - 3x_3 = 0\) leads to \(x_1 = -x_3\). By setting \(x_3 = t\), where \(t\) is a parameter, we express the general solution as:

• \(x_1 = -t\)
• \(x_2 = -t\)
• \(x_3 = t\)

This gives \( (x_1, x_2, x_3) = t(-1, -1, 1) \), which indicates that every solution is a scalar multiple of the vector \((-1, -1, 1)\). This general solution captures the entire set of solutions for the given linear system, providing a comprehensive understanding of the vectors that fit the criteria.

In three-dimensional space \(R^3\), the solution space of a system of equations can describe various geometric objects, such as points, lines, or planes. In this scenario, the vector equation \( (x_1, x_2, x_3) = t(-1, -1, 1) \) describes a line passing through the origin of the coordinate system in \(R^3\).

• The parameter \(t\) allows the vector to extend infinitely in both directions on the line.
• The line only includes points where solutions are orthogonal to both \(\mathbf{a}\) and \(\mathbf{b}\), fulfilling the orthogonality condition.

Furthermore, according to Theorem 3.4.3, we have confirmed that the solution space must be a subspace in \(R^3\), which indeed a line through the origin is. Understanding these geometric implications helps visualize the solution space in a three-dimensional context, bridging algebraic solutions with geometric interpretations.