How Do You Visualize a Vector Field Like v(x, y) = (2.5, -x)?

[hexa]

Please could someone explain to me how to visualise a vector field? Let's say it's v(x, y) = (2.5, -x) on whatever domain. I tried it the same way as I would visualize a scalar field but the results did not correspondent at all with the results I'd expect.

The same for drawing the field along the x and y-axis and along lines through the middle that make a 45 degree angle with x and y. Again my results look somewhat weired...

Thanks a lot,
hexa

 

[HallsofIvy]

If v= (2.5, -x) then the x-component is 2.5 while the y component is -x. Every such vector is constant along a vertical x= a line.

Along the y-axis, for example, all vectors are (-2.5,0), pointing parallel to the negative x-axis with length 2.5. Along the line x= 1, all vectors are (-2.5, -1), now pointing down and to the left. Along the line x= 2, all vectors are (-2.5,-2), pointing left but even more "down". Along the line x= -1, all vectors are (-2.5, 1) pointing up and to the left.

It's much easier to draw the field along "isoclines" where the vector is a constant (here x= constant which includes x=0, the y-axis) but if you must draw along the x-axis, you get vectors that always have -2.5 x-component but y-component larger and larger: Think of a vector starting of pointing straight back but swinging more and more downward as you move along the positive x axis. Of course, the vector swings more and more upward as you move left along the negative x-axis. As far as the lines y= x, y= -x, just look down to the x-axis. The vector at a point on either y= x or y= -x looks exactly like the vector on the x-axis directly below or above the given point.

 

[Architeuthis Dux]

Visualizing a vector field can be a challenging task, but there are a few steps you can follow to help understand and accurately depict the field.

First, it is important to understand the concept of a vector field. A vector field is a mathematical function that assigns a vector (direction and magnitude) to each point in a given space. In the case of your example, the vector field v(x, y) = (2.5, -x) assigns a vector with a magnitude of 2.5 and a direction of -x (negative x) to every point (x, y) in the domain.

To visualize this, you can plot the vectors at various points in the domain. For example, you can choose a few points (x, y) and plot a vector with a magnitude of 2.5 and a direction of -x at each point. This will give you a sense of the overall direction and magnitude of the vector field at those points.

Next, you can plot the field along the x and y-axis by assigning different values of x and y and plotting the corresponding vectors. This will give you a better understanding of how the vector field changes along these axes.

To plot the field along lines that make a 45 degree angle with the x and y-axis, you can use the same approach as before. Choose points along these lines and plot the vectors with the appropriate magnitude and direction.

It is important to note that the visualization of a vector field may not always correspond exactly with your expectations, especially if the field is complex. However, by following these steps and experimenting with different points and lines, you can gain a better understanding of the overall behavior of the vector field.