Problem 5 - Calculus Volume 3

Question

Describe each vector field by drawing some of its vectors. $$ \mathbf{F}(x, y)=x \mathbf{i}+y \mathbf{j} $$

Step-by-Step Solution

Answer

The vector field radiates outward, with longer vectors further from the origin, symmetric in all directions.

Step 1: Understanding the Vector Field

This vector field, \( \mathbf{F}(x, y) = x \mathbf{i} + y \mathbf{j} \), is defined in terms of its components in the \( x \) and \( y \) directions. For each point \( (x, y) \), the vector \( \mathbf{F} \) has an \( x \)-component equal to \( x \) and a \( y \)-component equal to \( y \). This means at point \( (x, y) \), the vector points in the direction away from the origin and its length increases with both \( x \) and \( y \).

Step 2: Choosing Points to Plot Vectors

To effectively draw this vector field, select key points on the coordinate plane. Common choices might include points like \( (1,0) \), \( (0,1) \), \( (-1,0) \), \( (0,-1) \), \( (1,1) \), and \( (-1,-1) \). These points will help visualize the field's behavior in different quadrants and along the axes.

Step 3: Drawing Vectors

Draw vectors at the selected points. For example, at \((1, 0)\), the vector is \( \mathbf{i} \), pointing along the positive \( x \)-axis. At \((0, 1)\), the vector is \( \mathbf{j} \), pointing along the positive \( y \)-axis. At \((1, 1)\), the vector is \( \mathbf{i} + \mathbf{j} \), pointing diagonally outward (45 degrees above the \( x \)-axis). Use arrows to indicate direction and ensure that each vector starts at its corresponding point.

Step 4: Describing Vector Characteristics

Note that the vectors radiate outward from the origin and increase in magnitude as \( x \) and/or \( y \) increase. This shows that the field is uniform in all directions but increases linearly with distance from the origin. The vector field is symmetric about the origin.

Key Concepts

Vector Components

In vector calculus, vector components refer to the projection of a vector along the coordinate axes. Each vector in a plane can be broken down into two components: one along the x-axis and another along the y-axis. In our vector field example, the vector \( \mathbf{F}(x, y) = x \mathbf{i} + y \mathbf{j} \) consists of an x-component \( x \mathbf{i} \) and a y-component \( y \mathbf{j} \). This breakdown is useful because it allows us to simplify and analyze the vector in terms of its horizontal and vertical influences.

• The x-component (\( x \mathbf{i} \)) represents the effect of the vector in the horizontal direction, affected by the changes in the x-coordinate.
• The y-component (\( y \mathbf{j} \)) represents the vertical influence, depending on the changes in the y-coordinate.

Understanding these components helps in visualizing how vectors behave in different regions of the coordinate plane. It becomes clear that as one moves away from the origin, the length of each vector increases incrementally in response to changes in \( x \) and \( y \).

Coordinate Plane

The coordinate plane, crucial for vector analysis, is a two-dimensional surface on which we plot vectors based on their components. It's composed of the x-axis and y-axis, intersecting at the origin \((0,0)\), which serves as a central reference point.

• The positive x-axis extends to the right, while the negative x-axis extends to the left.
• Likewise, the positive y-axis extends upwards, and the negative y-axis extends downwards.

When examining a vector field such as \( \mathbf{F}(x, y) = x \mathbf{i} + y \mathbf{j} \), the coordinate plane is vital for understanding how vectors are distributed and how they interact across different quadrants. By selecting various points like \((1,0)\), \((0,1)\), and \((-1,-1)\), one can effectively visualize the vector field's characteristics and the symmetry around the origin. The coordinate plane provides the framework within which vectors are drawn.

Magnitude and Direction

The magnitude and direction of a vector describe its size and where it points. When considering the vector field \( \mathbf{F}(x, y) = x \mathbf{i} + y \mathbf{j} \), each vector's magnitude can be calculated using the Pythagorean theorem. The magnitude \( |\mathbf{F}| \) is given by \( \sqrt{x^2 + y^2} \).

• The greater the values of \( x \) and \( y \), the larger the magnitude, meaning the vector is longer and extends further from the origin.
• The direction is determined by the angle the vector makes with the x-axis, calculated using \( \tan^{-1}(\frac{y}{x}) \).

Understanding both these aspects allows us to predict how vectors behave in spaces of varying coordinate values. Vectors in this field point outward, away from the origin, and they become more pronounced (increase in magnitude) as you move further into the positive or negative regions of the plane. This knowledge is instrumental in fields like physics and engineering where vector decomposition is frequently used to analyze forces.

Symmetric Field

A symmetric field refers to a vector field that demonstrates uniformity and balance around a central point, often the origin in the coordinate plane. In our exercise, the vector field given by \( \mathbf{F}(x, y) = x \mathbf{i} + y \mathbf{j} \) is symmetrical about the origin.

• Each vector reflects similar behavior as counterpart vectors located symmetrically across the origin.
• For instance, vectors at \((1,1)\) and \((-1,-1)\) show symmetry because they have equal magnitude but point in opposite directions.

This symmetry means the field doesn't prefer any direction inherently, which makes analyzing it quite straightforward. If you were to draw vectors for this field, they would appear in a balanced pattern radiating outwards, forming a pattern that looks the same no matter which angle you view it from around the origin. Understanding symmetry in fields is useful for predicting vector behaviors and for simplifying complex mathematical problems often found in airflow, electromagnetism, and fluid dynamics.