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. \()\) \(\vec{F}=\langle 0, x\rangle\)

Short answer

Sketch vertical vectors at integer points with length equal to their x-coordinate.

Step-by-step solution

Understanding the vector field

The vector field is given by \( \vec{F} = \langle 0, x \rangle \). This means that at every point \((x, y)\), the vector has an x-component of 0 and a y-component equal to the x-coordinate of the point.

Identifying integer coordinate points

We are asked to sketch vectors at every point where both coordinates are integers within the given rectangle. The rectangle's corners are \((-2, -2)\) and \((2, 2)\), so integer coordinates range from \(-2\) to \(2\) for both x and y.

Calculating vectors at integer points

For each point \((x, y)\), the vector will be \(\langle 0, x \rangle\). For instance, at \((0,0)\), the vector is \(\langle 0, 0 \rangle\); at \((1,2)\), the vector is \(\langle 0, 1 \rangle\); and at \((-2,-1)\), the vector is \(\langle 0, -2 \rangle\).

Sketching vectors

Draw the vectors on a coordinate plane with the given rectangle's bounds. Each vector will be vertical (since the x-component is 0) and have a length equal to the x-coordinate.

Key concepts

Coordinate Geometry

Coordinate geometry combines algebra and geometry to study the positions of points, lines, and shapes in a coordinate plane. This mathematical field uses a system of two perpendicular lines, known as axes, to define positions in space. The horizontal line is the x-axis, and the vertical line is the y-axis. Together, they form the Cartesian coordinate system.

In this system, every point is defined by an ordered pair of numbers \(x, y\), called coordinates, where 'x' represents the position on the x-axis and 'y' represents the position on the y-axis. Using coordinate geometry allows for precise definitions of geometric figures through algebraic equations. For example, the vector field \(\vec{F} = \langle 0, x \rangle\) specifies that at each point on the plane, the vector's direction depends on the x-coordinate alone. Understanding how coordinate systems function is crucial to working within vector fields and sketching them accurately.

Integer Coordinates

In mathematics, integer coordinates are points in space where both x and y are integers. They are whole numbers and do not include fractions or decimals. Working with integer coordinates is often simpler and more intuitive, making it easier to plot and analyze.

In the context of a vector field, evaluating vectors at integer coordinates means only considering points like (-2, -1), (0, 0), or (2, 1), where both x and y components are integers. This becomes particularly useful when trying to simplify calculations or visualizations. For the rectangle defined by corners (-2, -2) and (2, 2), integer coordinates range between these limits, resulting in a grid of whole-number points inside the rectangle. Sketched vectors rely on these coordinates to represent both direction and magnitude at regular intervals, providing a clear representation of the vector field.

Vector Components

Vectors consist of two main components: magnitude and direction. Expressed in coordinate form, a vector often appears as an ordered pair or triplet, for example, \(\langle x, y \rangle\) in 2D space. Each component represents part of the vector's effect in their respective axes.

• X-Component: The effect in the horizontal direction.
• Y-Component: The effect in the vertical direction.

In the vector field given by \(\vec{F} = \langle 0, x \rangle\), the x-component is always zero, indicating no horizontal movement, while the y-component mirrors the x-coordinate of the position on the plane. This setup results in vertical vectors of length proportional to the x-coordinate. Understanding these components helps in sketching and visualizing the field correctly, as it highlights the vertical nature of all vectors across the defined space.

Sketching Techniques

Sketching techniques for vector fields involve converting mathematical expressions into visual representations. A step-by-step process helps break down how this is achieved using both systematic calculations and creative visualization.

1. **Identify Coordinate Range:** Understand the region where the vectors will be plotted. In this exercise, the range is limited to integer coordinates within a rectangle bounded by (-2, -2) and (2, 2).

• Plot points where x and y are integers, creating a grid.

2. **Calculate Vector for Each Point:** For each integer coordinate point, compute the vector’s components using the field equation \(\vec{F} = \langle 0, x \rangle\).

• Keep in mind the x-component is zero, making all vectors vertical.

3. **Draw Vectors:** At each plotted point, draw the corresponding vector. The direction of the vector is along the y-axis, with length determined by the x-coordinate.

By following these techniques, a clear, organized depiction of the vector field is developed. This approach ensures all vectors are drawn with consistency and accuracy, reflecting both the magnitude and direction designated by the vector field equation.