Problem 3 - Advanced Engineering Mathematics

Question

In Problems \(1-6\), graph some representative vectors in the given vector field. $$ \mathbf{F}(x, y)=y \mathbf{i}+x \mathbf{j} $$

Step-by-Step Solution

Answer

The vector field represents rotational patterns centered at the origin.

Step 1: Analyze the Vector Field Formula

The given vector field \( \mathbf{F}(x, y) = y\mathbf{i} + x\mathbf{j} \) indicates that each vector \( \mathbf{F} \) at a point \((x, y)\) is given by \( y \) in the \( x \)-direction (\( \mathbf{i} \)) and by \( x \) in the \( y \)-direction (\( \mathbf{j} \)). This means that for each point \((x, y)\), the vector's x-component is \( y \) and the y-component is \( x \).

Step 2: Choose Representative Points

Select a few points in the plane to evaluate the vector field. For simplicity, start with points like \((1, 0)\), \((0, 1)\), \((1, 1)\), \((1, -1)\), \((2, 0)\), etc. These points should adequately cover different parts of the plane to understand the vector field's behavior.

Step 3: Calculate Vectors at Chosen Points

For each point, compute the vector \( \mathbf{F}(x, y) \): - At \((1, 0)\), \( \mathbf{F}(1, 0) = 0\mathbf{i} + 1\mathbf{j} = \mathbf{j} \).- At \((0, 1)\), \( \mathbf{F}(0, 1) = 1\mathbf{i} + 0\mathbf{j} = \mathbf{i} \).- At \((1, 1)\), \( \mathbf{F}(1, 1) = 1\mathbf{i} + 1\mathbf{j} \).- At \((1, -1)\), \( \mathbf{F}(1, -1) = -1\mathbf{i} + 1\mathbf{j} \).- At \((2, 0)\), \( \mathbf{F}(2, 0) = 0\mathbf{i} + 2\mathbf{j} = 2\mathbf{j} \).

Step 4: Graph the Vectors

Graph each calculated vector originating from its corresponding point, plotting their directions and magnitudes:- From point \((1, 0)\), draw a vector pointing upwards.- From point \((0, 1)\), draw a vector pointing rightwards.- From point \((1, 1)\), draw a vector pointing diagonally in the \( \mathbf{i} + \mathbf{j} \) direction.- From point \((1, -1)\), draw a vector pointing diagonally downleft in the \( -\mathbf{i} + \mathbf{j} \) direction.- From point \((2, 0)\), draw a vector twice as long as the one at \((1, 0)\) pointing upwards.

Step 5: Interpret the Vector Field

Notice the pattern in the vector directions: vectors in the field indicate a rotational or circular motion around the origin. Each vector at a point \((x, y)\) suggests that as either \(x\) or \(y\) increases, the vector's magnitude generally corresponds to their coordinate values, reflecting a pattern of rotation.

Key Concepts

Vector Components

Vectors are fundamental components in the description of vector fields. In a vector field, each point in space has a vector associated with it. This vector is broken down into two components along the horizontal and vertical axes. The components are commonly represented using the unit vectors \( \mathbf{i} \) for the x-axis and \( \mathbf{j} \) for the y-axis.
In the given vector field formula \( \mathbf{F}(x, y) = y\mathbf{i} + x\mathbf{j} \), each vector \( \mathbf{F} \) at any specific point \((x, y)\) is described by:

• The x-component which is represented by the magnitude \( y \) in the direction of \( \mathbf{i} \)
• The y-component which is represented by the magnitude \( x \) in the direction of \( \mathbf{j} \)

This means the vector acts in the direction along each axis proportional to the other coordinate. This interplay of components forms the basis for constructing and understanding the field's directional patterns.

Graphing Vectors

Graphing vectors in a vector field provides a visual understanding of how vectors are distributed across the field. To graph vectors, follow these steps:
First, select representative points on your coordinate plane to fully understand how vectors behave in the space. In our solution, we chose points such as \((1, 0), (0, 1), (1, 1)\), and so on. For each point, calculate the vector using the field formula. Graph each vector starting from its selected point, ensuring its direction and magnitude are reflected appropriately.
This might include:

• Graphing a vector pointing upwards at \((1, 0)\)
• Drawing a rightward pointing vector at \((0, 1)\)
• Depicting diagonal vectors which combine horizontal and vertical movements at points like \((1, 1)\)

Through graphing, the geometric patterns of vector fields, like circular or rotational motions, become clear. Graphing lays out how vectors interrelate and interact across the coordinate plane.

Rotational Motion

In observing a vector field like \( \mathbf{F}(x, y) = y\mathbf{i} + x\mathbf{j} \), you can identify patterns such as rotational motion. The design of the field indicates that vectors encircle around the origin as a central point.
To identify rotational motion, observe the direction vectors take as coordinates change.

• As you draw vectors, notice whether they turn around the origin in consistent pathways.
• Observe the change in direction and magnitude; as they spiral around a point, it indicates rotation.

Each vector represents force vectors being different in each location but forming a pattern of rotational flow. This is often described visually as forming circles or segments of a spiral, highlighting how powerful vector algebra can be in studying physics and engineering.

Coordinate System

Understanding the coordinate system is essential for graphing and analyzing vector fields. In our case, we work with a 2D Cartesian coordinate system, which consists of two perpendicular axes—the x-axis and the y-axis. Each point in this system is represented as \((x, y)\), determining a unique location on the plane.
Every vector in our vector field is affected by its position within this coordinate space:

• The x-axis, aligned horizontally, influences how the y-component of vectors appear.
• The y-axis, aligned vertically, affects the x-component of vectors.

This relationship shows how motion and direction are interpreted differently depending on the coordinates. When graphing, it's essential to correctly place each vector at its respective point and account for how the axes influence its components. This enhances our comprehension of how the mathematical constructs govern motion and direction in the field.