91Ó°ÊÓ

Understanding vector equations is crucial because they allow us to express relationships between vectors just like regular algebraic equations do with scalars. In vector equations, vectors are often represented in bold, such as \( \mathbf{u} \) and \( \mathbf{v} \) in our exercise.

A vector equation can involve operations like addition, subtraction, or scalar multiplication of vectors, resulting in another vector. For example, in the exercise, we have the equation:

\[ 2 \mathbf{u} + 3 \mathbf{v} = \mathbf{i} \]

Here, the equation describes a linear combination of vectors \( \mathbf{u} \) and \( \mathbf{v} \) that equals vector \( \mathbf{i} \). Solving vector equations often involves breaking the vectors into components, and treating each component individually. This will transform the vector equation into a pair of algebraic equations, which are much easier to solve.

When dealing with vectors, it can be useful to express them in terms of their components, referred to as coordinate vectors. This involves specifying each vector in a list of numbers, corresponding to different dimensions or axes, such as \( \langle x, y \rangle \) for 2D vectors.

In our exercise, the vectors \( \mathbf{i} \) and \( \mathbf{j} \) serve as the basis vectors:

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

These are unit vectors, pointing along the x and y axes, respectively. A vector \( \mathbf{a} \) can be expressed using these as \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \), where \( a_1 \) and \( a_2 \) are scalars.

By assigning coordinates to vectors, it becomes straightforward to perform operations like addition and subtraction, as in the step-by-step solution where the equations are rewritten in coordinate form. This method simplifies solving equations by translating them into problems involving numbers that can be managed via linear algebraic manipulations.

A linear combination involves creating a new vector by adding together a set of vectors multiplied by specific scalars. In the context of our exercise, the equation \( 2 \mathbf{u} + 3 \mathbf{v} = \mathbf{i} \) is a linear combination of the vectors \( \mathbf{u} \) and \( \mathbf{v} \).

• The number '2' is the scalar multiplying vector \( \mathbf{u} \).
• The number '3' scales vector \( \mathbf{v} \).

Finding solutions for such linear combinations usually requires expressing one vector in terms of the others, as shown in the step-by-step solution where \( \mathbf{u} \) was expressed in terms of \( \mathbf{v} \).

Linear combinations are foundational in vector spaces, where they help determine vectors' dependencies and span. By understanding how vectors can combine, we gain insights into their interactions, enabling us to solve complex equations and understand geometric interpretations.