91Ó°ÊÓ

When working with vectors in a plane, understanding the coordinate axes is fundamental. The coordinate axes consist of the x-axis and the y-axis, which are perpendicular to one another. These axes allow us to determine the position of any point in the plane by using two numerical coordinates: the x-coordinate, which reflects the horizontal position, and the y-coordinate, which indicates the vertical position.

• The point where the x-axis and y-axis intersect is called the origin, represented by \( (0, 0) \).
• Vectors are often drawn starting from the origin.
• In our specific exercise, by plotting the vectors \( \mathbf{u} \), \( \mathbf{v} \), and \( \mathbf{w} \), we can visually analyze their positions in relation to the coordinate axes.

In terms of linear combinations, the coordinate axes help us visualize how different vectors can be scaled and added to reach a specific point or vector in the plane. By sketching \( \mathbf{u} \) and \( \mathbf{v} \) starting from the origin, you can easily see how they add up together to form another point, in this case, \( \mathbf{w} \). Relying on these foundational axes helps in understanding the system of equations that arise from expressing one vector as a combination of others.

A system of equations is a set of two or more equations with the same set of unknowns. In linear algebra, finding a linear combination of vectors often involves solving a system of linear equations. In our exercise, we were given the equation \( a\mathbf{u} + b\mathbf{v} = \mathbf{w} \), which composed a system of equations as follows:

• \( -2a + 2b = 2 \)
• \( 3a + b = 9 \)

To find the values of \( a \) and \( b \), we solved these equations using substitution and elimination methods. Here's a brief summary of these methods:

• **Substitution Method:** Solve one of the equations for one variable and substitute this expression into the other equation. This reduces the system to a single equation with one unknown, which can be solved directly.
• **Elimination Method:** Add or subtract equations to eliminate one of the variables, making it easier to solve for the remaining unknown.

In our solution, we simplified the first equation to find that \( b = a + 1 \) and substituted into the second equation, which allowed us to solve for \( a \), finding \( a = 2 \). Using this value, \( b \) was then easily found to be 3.
By solving this system of equations, we concluded that \( \mathbf{w} \) is appropriately expressed as the linear combination \( 2\mathbf{u} + 3\mathbf{v} \).

Vectors are essential in representing quantities that have both magnitude and direction, making them incredibly useful in physics and engineering. In mathematics, vectors are used in various applications such as representing points in a plane and conducting operations like addition and scalar multiplication. In our exercise:

• \( \mathbf{u} \) has components \( -2 \) and \( 3 \), which means it moves 2 units to the left and 3 units upward.
• \( \mathbf{v} \) has components \( 2 \) and \( 1 \), indicating movement 2 units to the right and 1 unit upward.
• \( \mathbf{w} \) is given as \( 2 \) and \( 9 \), and we explore how it can be represented as a combination of the two vectors.

Using linear combinations of vectors involves scaling vectors by real numbers (scalars) and then summing them to find a specific vector result.

• In this context, expressing \( \mathbf{w} \) as \( 2\mathbf{u} + 3\mathbf{v} \) illustrates the principle of linear combinations, as it highlights how multiple vectors can interact to form a new one.
• Understanding how vectors interact can lead to deeper insights in fields such as linear transformations and vector spaces.

Grasping how vectors work and their properties equips you to handle complex mathematical problems, making them a pivotal concept in advanced mathematics and physics.