Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-6, sketch a sample of vectors for the given vector field \(\mathbf{F}\). $$ \mathbf{F}(x, y)=x \mathbf{i}+y \mathbf{j} $$

Step-by-Step Solution

Answer

Plot vectors at various points; they point radially from the origin.

Step 1: Understand the Vector Field Components

The vector field \( \mathbf{F}(x, y) = x \mathbf{i} + y \mathbf{j} \) can be broken down into components. The \( x \)-component is represented by \( x \mathbf{i} \), and the \( y \)-component is represented by \( y \mathbf{j} \). This means for any point \( (x, y) \), the vector \( \mathbf{F} \) has components \( x \) in the \( i \)-direction (horizontal) and \( y \) in the \( j \)-direction (vertical).

Step 2: Choose Specific Points

Select specific points on the plane to plot vectors. Common points to start with might include \( (0, 0), (1, 0), (0, 1), (1, 1), (-1, 0), (0, -1), (-1, -1) \), and more. Substitute these points into \( \mathbf{F}(x, y) \) to find the vector associated with each point.

Step 3: Compute Vector Components

For each selected point, compute the vector by substituting the \( x \) and \( y \) values into the expression \( \mathbf{F}(x, y) = x \mathbf{i} + y \mathbf{j} \). For instance, for point \( (1, 0) \), the vector is \( 1 \mathbf{i} + 0 \mathbf{j} \), which is simply \( \mathbf{i} \).

Step 4: Sketch the Vectors

Plot each vector starting from its corresponding point. Use arrows to represent each vector's direction and magnitude. Remember that vectors should originate from their initial points; for example, plot the vector \( \mathbf{i} \) starting at \( (1, 0) \). All arrows should be drawn proportionally to their magnitude.

Step 5: Recognize the Vector Field Pattern

Notice that the vector at each point \( (x, y) \) is pointing directly towards the position \( (x, y) \). This means vectors in this vector field point radially outwards from the origin, with magnitude increasing as you move away from the origin. This indicates a linear relationship between a vector's length and its distance from the origin.

Key Concepts

Vector Components

Vector components are crucial in understanding how vectors function within a field. In any vector field, like \( \mathbf{F}(x, y) = x \mathbf{i} + y \mathbf{j} \), each vector is an expression of two components, namely the \( x \)-component and the \( y \)-component. These components indicate the direction and magnitude along the respective axes.
To break it down:

• The \( x \)-component is represented by \( x \mathbf{i} \), denoting movement along the horizontal axis.
• The \( y \)-component is represented by \( y \mathbf{j} \), denoting movement along the vertical axis.

Each of these components gives the vector a unique position in the vector field. By identifying these components at any point \((x, y)\), we can determine the vector's direction and how far it stretches in each direction.

Plotting Vectors

Plotting vectors helps visualize and understand the behavior of vector fields. When plotting vectors from \( \mathbf{F}(x, y) = x \mathbf{i} + y \mathbf{j} \), begin by selecting specific points, such as \((0,0)\), \((1,1)\), or \((-1,-1)\).
Follow these steps:

• For each selected point, compute the vector by inserting the \( x \) and \( y \) values into the equation \( \mathbf{F}(x, y) \).
• Draw the vector starting at the selected point with an arrow indicating direction.
• Remember, the length of the arrow should be proportional to the vector's magnitude.

Plotting vectors will show how they behave around each other, displaying the overall flow and behavior of the vector field.

Vector Magnitude

The magnitude of a vector defines how long a vector is and gives insight into the field's intensity at any point. For vector \( \mathbf{F}(x, y) = x \mathbf{i} + y \mathbf{j} \), the magnitude can be calculated using the formula:\[\text{Magnitude} = \sqrt{x^2 + y^2}\]
Here's what you need to keep in mind:

• The magnitude is always a positive value, providing the length of the vector.
• A larger magnitude signifies that the vector stretches further from the origin.
• The vector's magnitude grows as you move farther from the origin in this field.

Understanding this concept helps in estimating the force or effect a vector represents in various parts of the vector field.

Radial Vectors

Radial vectors are vectors that point directly away from or towards a center point, often the origin in a vector field. In the case of the vector field \( \mathbf{F}(x, y) = x \mathbf{i} + y \mathbf{j} \), these vectors are radial because:

• Each vector points outward from the origin \((0, 0)\), aligning with the point's coordinates \((x, y)\).
• The direction of each vector is determined by its position along the axes \( \mathbf{i} \) and \( \mathbf{j} \).
• The magnitude of each radial vector increases as the distance from the origin increases.

Understanding radial vectors is essential for recognizing the structure of the vector field. It reflects how vectors spread uniformly from the origin, almost like rays emanating from the sun.