Question

$$ \mathbf{F}(x, y)=(2 y-3 x) \mathbf{i}+(2 y+3 x) \mathbf{j} $$

Short answer

The vector field \( \mathbf{F}(x, y)=(2 y-3 x) \mathbf{i}+(2 y+3 x) \mathbf{j} \) has components that point towards the Y-axis and -X axis as x and y increase, respectively for the i-component and towards the X and Y axes as y and x increase, respectively for the j-component.

Step-by-step solution

Understand the Vector Field

Examine the vector field \( \mathbf{F}(x, y)=(2 y-3 x) \mathbf{i}+(2 y+3 x) \mathbf{j} \). It is seen that there are two components for this vector field – one along the X-axis and the other along the Y-axis. These are represented by coefficients of the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\), respectively.

Examine the Component along the X-axis

The component of the vector field along the X-axis is given by \( (2 y-3 x) \mathbf{i} \). This represents a vector that tends towards the Y-axis as x increases and towards the -X axis as y increases.

Examine the Component along the Y-axis

The component of the vector field along the Y-axis is given by \( (2 y+3 x) \mathbf{j} \). This represents a vector that tends towards the X-axis as y increases and towards the Y axis as x increases.

Key concepts

Unit Vectors

In understanding vector fields, it’s vital to start with the basics of unit vectors. Unit vectors are vectors with a magnitude of 1, and they are used to provide direction without scaling. They are the foundation upon which vector components and vector fields are built. The most common unit vectors in two-dimensional space are \( \mathbf{i} \) and \( \mathbf{j} \) which point along the X-axis and Y-axis, respectively.

In the given exercise, \( \mathbf{i} \) and \( \mathbf{j} \) are used to describe the vector field \( \mathbf{F}(x, y) \) in terms of its horizontal (\( x \) direction) and vertical (\( y \) direction) components. Since they are unit vectors, they define the direction of the vector field without affecting its magnitude caused by the scalar components.

Vector Components

Vector components are the building blocks of a vector, projecting the vector onto the different axes of the coordinate system. In the context of our exercise, the vector field \( \mathbf{F} \) consists of two primary components: \( (2y-3x) \mathbf{i} \) and \( (2y+3x) \mathbf{j} \) which correspond to the horizontal and vertical components of the field respectively.

It is important to note that each component affects how the vector \( \mathbf{F} \) behaves. For example, a positive component along \(\mathbf{i} \) indicates a vector pointing rightward along the X-axis, while a negative value suggests the opposite direction. Similarly, the \(\mathbf{j} \) component dictates the up-and-down movement along the Y-axis. By understanding the components of a vector, students can visualize the combined effects these components have on the overall direction and magnitude of a vector within a field.

Calculus

Calculus is a branch of mathematics that deals with rates of change and accumulation. In relation to vector fields and their components, calculus allows us to explore how these fields change over space. It's crucial for students to grasp that vector fields like \( \mathbf{F}(x, y) \) can depict the flow of a fluid or the force exerted on an object at every point in space, and calculus provides the tools to analyze such complex behavior.

One key aspect of calculus in vector fields is the gradient, which tells us how a scalar field varies and hence, helps in computing the direction and magnitude of the maximum rate of increase. Another important concept is the divergence of a vector field, which measures the extent to which the vector field flows outward from a point. In advanced studies, concepts like curl and line integrals of vector fields become significant, demonstrating how calculus is indispensable for understanding the dynamics of physical phenomena represented by vector fields.