Problem 6 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

Graph vectors have only vertical components \( x \mathbf{j} \), longer for larger \( |x| \).

Step 1: Understand the Vector Field

In this problem, we are given the vector field \( \mathbf{F}(x, y) = x \mathbf{j} \). This expression suggests that for each point \((x, y)\) in the plane, the vector points in the positive or negative \( y \)-direction (along the \( j \) axis) and its magnitude is equal to the \( x \)-coordinate of the point.

Step 2: Break Down the Vector Components

The vector field \( \mathbf{F}(x, y) = x \mathbf{j} \) can be expressed in terms of its components as \( \langle 0, x \rangle \). This means at each point \((x, y)\), the vector has 0 component in the \( i \) direction and \( x \) component in the \( j \) direction.

Step 3: Select Points to Graph Vectors

To graph some representative vectors, choose several points in the \((x, y)\) plane. A good selection might be \((1, 0), (2, 0), (-1, 0), (1, 1), (0, 1)\). The \( x \) component determines the length of the vector in the \( y \)-direction for each of these points.

Step 4: Calculate and Draw Vectors

For each selected point:- At \((1,0)\), the vector is \( \langle 0, 1 \rangle \).- At \((2,0)\), the vector is \( \langle 0, 2 \rangle \).- At \((-1,0)\), the vector is \( \langle 0, -1 \rangle \).- At \((1,1)\), the vector is \( \langle 0, 1 \rangle \).- At \((0,1)\), the vector is \( \langle 0, 0 \rangle \) (no vector).Draw these vectors on a graph with their respective lengths and directions.

Step 5: Analyze the Vector Field Pattern

Notice that vectors at different points align vertically (along the \( y \)-axis) and their magnitude depends solely on the \( x \) coordinate. This means vectors are longer for larger absolute values of \( x \) and zero when \( x = 0 \). This creates vertical lines of vectors of varying lengths depending on their \( x \) position.

Key Concepts

Vector Components

In understanding vector fields like the one given, it helps to break them down into their component parts. The vector field \( \mathbf{F}(x, y) = x \mathbf{j} \) can be visualized as vectors that only act in the vertical direction. When we talk about vector components, we are referring to how a vector can be split into parts that run parallel to the axes. In this case:

• The vector components are \( \langle 0, x \rangle \).
• The \( i \)-component, which would be the horizontal part, is zero.
• The \( j \)-component represents motion in the \( y \)-direction, equal to the \( x \)-coordinate value.

This means that at any point \((x, y)\), the vector only moves vertically (up or down depending on the sign of \(x\)), showcasing the importance of understanding vector components. It's like tearing a vector apart and analyzing its individual impacts on each direction in the plane.

Graphing Vectors

Graphing vectors from a vector field involves choosing specific points and drawing the corresponding vectors based on their components. This process allows you to visualize how the vector field affects the plane.

• Select key points on the plane; in this example, points such as \((1, 0)\), \((2, 0)\), and \((1, 1)\) are effective for illustrating the field.
• Calculate the vector at each point. For instance, at \((1,0)\), the vector is \(\langle 0, 1 \rangle \) as it extends 1 unit upwards along the \( y \)-axis.
• Use arrows to represent vectors, indicating both direction and magnitude.

The choice of points should illustrate the general behavior across the plane. By drawing vectors at both positive and negative \(x\) values, you show how direction changes depending on the position in the field, providing a comprehensive picture of the vector field's effect.

Plane Vectors

Plane vectors are essential for understanding the dynamics at play in two-dimensional spaces like the \((x, y)\) plane. Here, vectors lie entirely within the plane and can be conceptualized visually:

• Vectors in this plane are described by \( \langle x, y \rangle \), indicating horizontal and vertical components respectively.
• However, in the given field \( \mathbf{F}(x, y) = x \mathbf{j} \), all vectors are zero in the \(i\) direction, so they only have vertical components.
• These vectors can stretch, shrink, or flip direction based solely on their \(x\) position, emphasizing how plane vectors adapt to field changes.

When analyzing a vector field on a plane, understanding these plane vectors and their components is crucial. It helps to predict the movement and behavior of objects under the influence of such fields, making them valuable in physics and engineering applications.