Question

Sketch the vector field \(\mathbf{F}=\langle x, y\rangle.\)

Short answer

Answer: In the vector field \(\mathbf{F}=\langle x, y\rangle\), the vectors point away from the origin and their magnitude increases as we move away from the origin, representing a radial outward flow.

Step-by-step solution

Understanding the Vector Field Components

The given vector field is \(\mathbf{F}=\langle x, y\rangle\). This means that at each point \((x, y)\) in the x-y plane, the vector has its x-component equal to the x-coordinate of the point and its y-component equal to the y-coordinate of the point. In other words, the vector \(\mathbf{F}\) at a point \((x, y)\) is: $$\mathbf{F}(x, y) = \langle x, y\rangle$$

Selecting Points to Sketch the Vector Field

To sketch the vector field, we'll choose a set of points in the x-y plane (preferably covering all four quadrants) and plot the vector corresponding to each point. We will select some points of the form \((x, y)\) such as: $$(-2, -2), (-1, -1), (0, 0), (1, 1), (2, 2), (-2, 2), (-1, 1), (2, -2), (1, -1)$$

Plotting the Vectors

Now we will plot the vectors at each point that we selected in Step 2. The vector at a point \((x, y)\) is given by \(\mathbf{F}(x, y) = \langle x, y\rangle\). For example, at the point \((-2, -2)\), the vector will be \(\langle -2, -2 \rangle\). Similarly, we can calculate the vectors at all selected points. Once we plot the vectors, we observe a pattern in the vector field. The vectors are always pointing away from the origin, and their magnitude increases as we move away from the origin. Thus, the vector field represents a radial outward flow. The final sketch will show the vectors at the selected points, illustrating the behavior of the vector field \(\mathbf{F}=\langle x, y\rangle\).

Key concepts

vector components

An important aspect of vector fields is understanding their vector components. In this exercise, the vector field \(\mathbf{F} = \langle x, y \rangle\) is specified. This notation means that at each point \((x, y)\) in the plane, the vector \(\mathbf{F}\) has an x-component and a y-component.

• The x-component of the vector corresponds to the x-coordinate of the point.
• The y-component of the vector corresponds to the y-coordinate of the point.

Therefore, if we have a point such as \((1, 2)\), the vector at this point is \(\langle 1, 2 \rangle\). This means the vector is directed one unit in the x-direction and two units in the y-direction from the point located at the origin \((0, 0)\).
Breaking down vectors into these components helps us to visualize and sketch the vector field more effectively, as it simplifies identifying direction and magnitude at any given point.

x-y plane

The x-y plane is the two-dimensional space we typically use to represent vectors like \(\mathbf{F} = \langle x, y \rangle\). This plane is made up of two perpendicular axes: the x-axis and the y-axis. In this context, every point on the plane can be represented as a coordinate pair \((x, y)\).
To visualize the vector field, it is useful to select various points on this plane. These points commonly cover all four quadrants:

• Quadrant I: both x and y are positive, e.g., \((1, 1)\).
• Quadrant II: x is negative, y is positive, e.g., \((-1, 1)\).
• Quadrant III: both x and y are negative, e.g., \((-1, -1)\).
• Quadrant IV: x is positive, y is negative, e.g., \((1, -1)\).

Each vector will originate from its corresponding point in these quadrants, and the overall pattern of these vectors illustrates the behavior of the vector field. Selecting points across the plane allows us to have a complete picture of how vectors behave at different locations.

radial outward flow

In the given vector field \(\mathbf{F} = \langle x, y \rangle\), the vectors exhibit a radial outward flow. This means all the vectors are directed away from a central point — in this case, the origin \((0, 0)\).

• The direction of each vector at any point \((x, y)\) is directly outward from the origin.
• The magnitude of the vectors increases with the distance from the origin.

For instance, at point \((2, 2)\), the vector \(\langle 2, 2 \rangle\) is longer than the vector at \((1, 1)\), which is \(\langle 1, 1 \rangle\). This increase in length signifies that the farther a point is from the origin, the stronger the vector is in pushing away from the center.
This characteristic—radiality—is what creates a visual pattern in the sketch of the vector field, where all vectors seem to stream outward symmetrically from the center, resembling a starburst pattern. Understanding radial outward flow is key to interpreting vector fields and predicting how they influence movement or distribution in a given space.