Problem 4 - Advanced Engineering Mathematics

Question

Graph some representative vectors in the given vector field. $$ \mathbf{F}(x, y)=x \mathbf{i}+2 y \mathbf{j} $$

Step-by-Step Solution

Answer

Graph vectors such as \((0,0)\), \((1,1)\), \((-1,1)\), etc., following their components \(x\mathbf{i} + 2y\mathbf{j}\).

Step 1: Understand the Vector Field

The given vector field \( \mathbf{F}(x, y) = x \mathbf{i} + 2y \mathbf{j} \) is a field of vectors where each vector depends on the position \( (x, y) \). The component \( x \mathbf{i} \) determines the magnitude and direction along the x-axis, while \( 2y \mathbf{j} \) determines the magnitude and direction along the y-axis.

Step 2: Choose Representative Points

Select several points in the plane to calculate and graph vectors. For instance, choose points like \((0, 0), (1, 1), (-1, 1), (1, -1), (2, 0)\), and so on.

Step 3: Calculate the Vector at Each Point

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

Step 4: Graph Each Calculated Vector

Using graph paper or a digital graphing tool, plot each vector starting from its corresponding point. The direction and length of the vector are determined by its \( \mathbf{i} \) and \( \mathbf{j} \) components.

Step 5: Analyze the Vector Field Pattern

Observe the vectors' direction and magnitude. Notice that the x-component depends linearly on \( x \), and the y-component depends linearly on \( y \), indicating a pattern where vectors tend to stretch out as \( x \) and \( y \) increase.

Key Concepts

Vector Components

To truly understand vector fields, you need to grasp the concept of vector components. Vectors are entities that have both a magnitude and a direction. In our given vector field, \( \mathbf{F}(x, y) = x \mathbf{i} + 2y \mathbf{j} \), each vector can be broken down into two main components:

• \( x \mathbf{i} \): This denotes the horizontal (or x-axis) component of the vector. It tells us how far right or left the vector will point on the graph, depending on the value of \( x \).
• \( 2y \mathbf{j} \): This denotes the vertical (or y-axis) component. It shows us how far up or down from the base point the vector will go based on twice the value of \( y \).

Understanding these components is crucial as they allow you to visualize how the vector behaves at different points within the vector field.Each component indicates how the vector stretches or contracts along the x and y directions, respectively. By calculating vectors at multiple points, patterns such as parallel movements or perpendicular angles can become apparent.

Graphing Vectors

Graphing vectors helps in visualizing how a vector field behaves at various points in a plane. To graph vectors from the vector field \( \mathbf{F}(x, y) \), follow these steps:

• Start by choosing some representative points in the plane. Typical points like \((0, 0), (1, 1), (-1, 1)\), etc. give a good spread across the graph.
• Calculate the specific vector for each point you picked. For instance, at \((1, 1)\), the vector is \( \mathbf{F}(1, 1) = 1 \mathbf{i} + 2 \mathbf{j} \).
• Plot each vector starting from the point you chose. This means placing the tail of the vector at the chosen point and using the calculated components \( x \mathbf{i} \) and \( 2y \mathbf{j} \) to find the head of the vector.

When plotting, remember that the direction an arrow points shows the direction of the vector. Its length indicates the vector's magnitude.Consistently following this procedure produces a clear graphical representation of the vector field. It reveals how vectors interact with one another and may highlight symmetries or trends moving across the graph.

Vector Magnitude and Direction

Each vector within our vector field \( \mathbf{F}(x, y) = x \mathbf{i} + 2y \mathbf{j} \) possesses its own magnitude and direction. The magnitude of a vector is akin to its total length or size. This can be calculated using the formula:\[\text{Magnitude} = \sqrt{(x)^2 + (2y)^2}\]The direction of a vector shows the route or angle it takes from its starting point. This direction can be determined by considering the angle \( \theta \) it makes with the positive x-axis:\[\theta = \tan^{-1}\left(\frac{2y}{x}\right)\]Together, magnitude and direction form a complete picture of the vector's presence in the field:

• Magnitude gives you an idea of how forceful or weak a vector is. Larger magnitudes tell of stronger vectors.
• Direction indicates where the vector is headed. Using angle calculations, you can easily illustrate this on a graph.

These qualities are vital for understanding the nature and dynamics of vector fields, especially when predicting or analyzing physical phenomena influenced by these vectors.