Vector fields are a way to describe the direction and magnitude of a flow in a given space. A vector field is defined as a collection of vectors that are associated with each point in space, which are then associated with a specific value.
These vectors can be associated with magnitudes and directions of flows, accelerations, or any other properties that define a flow. The strength of the flow and the direction it goes in is described by these vector fields.
Visualizing these vector fields is an important step in understanding them. Luckily, there are several ways to visualize a vector field depending on what aspect of the field you want to focus on. The first two visualizations will be discussed in this article.
The first type of visualization focuses on the values that comprise the vector field. These values can be represented as colors or shaded areas depending on the dimension of space the vectors are in.
Field lines run from the y-axis to the x-axis
The direction that the field lines run in is the direction that the vector field pushes particles in. Particles moving with the flow of the vector field would just move in the field lines, so they would be moving in the direction of x or y.
To see this more clearly, look at how dust and other particles move in a cyclone. The dust and other particles move up and to the right, because that is where the force of the wind pushes them.
Field lines always start and end at either the x-axis or y-axis, never at a point. If you drew a line starting at either axis, and extended it indefinitely, then you would have drawn every field line.
The vector field is the same as X + Y
In this example, the vector field is X + Y, where X is the x-axis and Y is the y-axis. All of the vectors in this example are drawn on the x-y plane, so all of the values for X and Y are 0.
The arrows show the direction of each vector, with blue indicating a positive vector and red indicating a negative one. You can see that all of these vectors point in different directions, which matches up with how all points on the circle move around.
This example shows you that if you draw all points on a circle as a vector field, then your vector field is simply the circle itself! Try drawing more complicated shapes to see this in action.
You can also try drawing these shapes yourself to test your understanding.
The vector field is −Y + X
In this example, the blogger shows several plots that are made with the same vector field. The blogger shows how to match the vector field F(x, Y) = X, −y with the correct plot by looking at the coordinate changes.
The first plot shows the x-coordinate staying constant while the y-coordinate decreases. The second plot shows the x-coordinate decreasing while the y-coordinate decreases. The third plot shows both coordinates increasing, which is not a valid vector field for this equation.
This is a good check to make sure you have the correct plot! Sometimes it is hard to distinguish between increasing and decreasing coordinates, so checking that the value of one coordinate changes in value while the other stays constant can help determine which are increasing and which are decreasing.
Field lines curve in a counterclockwise direction
The curve of the field lines is a key feature that distinguishes one field from another. As mentioned before, fields can exist in many dimensions, but for our everyday world, we consider just the three spatial dimensions and time.
Fields in physics can be thought of as a set of coordinates that describe where something might be located. For example, a magnetic field might describe where a magnet is located or where its influence is felt.
How does one recognize a curved field line? Look for vectors that point down and to the right. If you traced these vectors until they met the origin point, they would form a counterclockwise (left) circle.
Field lines curve in a clockwise direction
The next step is to determine the direction of the vector field lines. Field lines always curve in a clockwise direction, which can be determined by tracing the lines in a counterclockwise direction.
When tracing vector field lines in a counterclockwise direction, you will find that you reach the starting point before reaching the end point. This indicates that the field lines curve in a clockwise direction.
You can also tell if the field lines curve in a clockwise or counterclockwise direction by drawing a square around them. If the square has rounded corners, then it is curved in a clockwise direction. If it has sharp corners, then it is curved in a counter-clockwise direction.
Generalizing this concept, you can say that any vector field with curved corners in its plot has corner vectors that are pointing down, which means that the field line corner curves are pointing up (or vice versa).
F(x, −y) = X + Y
Now that we can match the vector field F with the correct plot, let’s talk about some more interesting vector fields.
When a function contains only x and y components, then the graph of the function is a plane. In other words, when only the x and y components of a vector field are present, then we have a 2-dimensional graph.
For example, if we take the vector field F(x, −y) = X + Y , then we can see that it is represented by a plane. The blue points on the graph represent points that are moving in the X direction, and the red points are moving in the −Y direction.
There are also some points that are moving in both directions at once. These points have an orange color to them because they are moving in both the X and Y directions.
F(x, −y) = −Y + X9) Lines are more closely packed near the origin10) Lines are more spread out near the origin
In this blog post, we will learn how to plot vector fields using two-dimensional (2-D) Cartesian coordinates. We will learn how to match the vectors with the correct plots and how to recognize which direction the flow is going in on each vector field.
To begin, we must first understand what a vector field is. A vector field is a set of vectors that have a magnitude and direction. These vectors represent the flow or movement of something. In this case, the something is either X (horizontal) or Y (vertical) coordinate values.
The first step is to organize your vectors in order from the origin (0, 0) and then label each one with an x or y value depending on its direction.
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