Question

Sketch the given vector field over the rectangle with opposite corners (-2,-2) and \((2,2),\) sketching one vector for every point with integer coordinates (i.e., at \((0,0),(1,2),\) etc. \()\) \(\vec{F}=\left\langle y^{2}, 1\right\rangle\)

Short answer

Sketch vectors \( \langle y^2, 1 \rangle \) at integer coordinates inside \((-2, -2)\) to \((2, 2)\).

Step-by-step solution

Understand the Vector Field

The vector field given is \( \vec{F} = \langle y^2, 1 \rangle \). This means that the \( x \)-component of the vector is \( y^2 \), and the \( y \)-component is a constant \( 1 \). This will affect how vectors are drawn at each point, as the direction and magnitude depend on \( y \).

Identify the Points

Identify all the integer coordinates within the rectangle with opposite corners \((-2, -2)\) and \((2, 2)\). These points are: \((-2, -2), (-2, -1), (-2, 0), (-2, 1), (-2, 2), (-1, -2), \ldots, (2, 2)\). There are 25 points in total.

Calculate and Draw Vectors

For each of the points, calculate the vector \( \vec{F} \) and draw it.- At \((-2, -2)\), the vector is \( \langle (-2)^2, 1 \rangle = \langle 4, 1 \rangle \).- At \((0, 1)\), the vector is \( \langle 1^2, 1 \rangle = \langle 1, 1 \rangle \).- At \((0, 0)\), the vector is \( \langle 0^2, 1 \rangle = \langle 0, 1 \rangle \).Repeat this calculation for each integer point and sketch the vectors with their tails at the respective points. The length and direction will vary depending on the calculated components.

Key concepts

Vector Components

A vector field consists of vectors that attach specific components to each point in space. In the field \(\vec{F} = \langle y^2, 1 \rangle \), the vectors are defined by two components: \(y^2\) which acts as the \(x\)-component, and \(1\), which remains a constant \(y\)-component. This means for any point \((x, y)\), the vector has an \(x\)-component that varies with and equals \(y^2\), while the \(y\)-component is always 1. Understanding these components is essential because they determine the direction and length of the vectors at any given point in the vector field.
The vector's length, also known as its magnitude, can be calculated with the formula:

• Magnitude = \( \sqrt{(y^2)^2 + 1^2} = \sqrt{y^4 + 1} \)

This equation shows how the length of each vector is influenced by the square of the \(y\)-coordinate of the point.

Integer Coordinates

Integer coordinates refer to points in a coordinate system where both the \(x\)-coordinates and \(y\)-coordinates are integers. In our scenario, we are considering a rectangle bound by the corners \((-2,-2)\) and \((2,2)\).
This region includes every point where both coordinates range from \(-2\) to \(2\). Specifically, here are all the integer coordinate points:

• Points on the x-axis: \((-2,0), (-1,0), (0,0), (1,0), (2,0)\)
• Points on the y-axis: \((0,-2), (0,-1), (0,1), (0,2)\)
• Points within the area: \((-2,-2), (-1,-2), (1,2), (2,1), etc.\)

There are a total of 25 points in total to consider, each adding a unique vector when plotted. Understanding integer coordinates helps simplify calculations as it gives discrete points to evaluate rather than a continuous range.

Plotting Vector Fields

Plotting a vector field involves representing the effect of the vector function at each determined point. For the vector field \(\vec{F} = \langle y^2, 1 \rangle \), each vector's tail is placed at the integer coordinates within our defined rectangle. Start by calculating the vector for each point by substituting the coordinates into the vector equation.
Follow these steps for plotting:

• Identify each point's location.
• Substitute the corresponding \(y\) value to find the \(x\)-component, \(y^2\).
• Always use \(1\) as the \(y\)-component.
• Determine the vector's direction and magnitude based on the components obtained.
• Draw the vector starting from its tail at the coordinate point.

This results in a collection of vectors, each uniquely describing the vector field's characteristic at every integer coordinate.

Graphical Representation of Vectors

Graphical representation in vector fields is a crucial part of visualizing complex mathematical relationships. It involves showing all the vectors associated with each point, illustrating both direction and magnitude effectively. Drawing these representations in the coordinate system transforms abstract mathematical expressions into something tangible.
When sketching our vector field, each vector starts from its corresponding integer coordinate point.

• The length of the vector is determined by its magnitude \(\sqrt{y^4 + 1}\).
• Direction is determined by the vector components, with the \(x\)-component being \(y^2\) and the \(y\)-component consistently \(1\).

A graphical vector field helps us see how vectors change direction and intensity across the field, enabling a better understanding of how they interact collectively. It brings clarity to the abstract concept by showing how the vector field functions are distributed over a particular area, such as the specified rectangle in this exercise.