Plotting Vector Field: V=(xi+yj+zk)/\sqrt{}(x^2+y^2+z^2)

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SUMMARY

The discussion focuses on plotting the vector field defined by the equation V=(xi+yj+zk)/√(x²+y²+z²). Participants emphasize the importance of selecting specific coordinates (x, y, z) to calculate and visualize the vector at those points. The vector's direction is determined solely by the components xi, yj, and zk, while the denominator √(x²+y²+z²) influences the vector's magnitude based on its distance from the origin. Understanding these elements is crucial for accurately representing the vector field on a graph.

PREREQUISITES
  • Understanding of vector fields and their components
  • Familiarity with 3D coordinate systems
  • Knowledge of mathematical notation for vectors
  • Experience with graphing tools or software for 3D plotting
NEXT STEPS
  • Research techniques for visualizing 3D vector fields using software like MATLAB or Python's Matplotlib
  • Learn about the implications of vector normalization in graphical representations
  • Explore the concept of divergence and curl in vector calculus
  • Investigate the effects of varying the distance from the origin on vector field magnitude
USEFUL FOR

Mathematicians, physicists, and engineers interested in visualizing and analyzing vector fields in three-dimensional space.

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How to plot this vector field on a graph
\stackrel{}{\rightarrow}
V=(xi+yj+zk)/\sqrt{}(x^2+y^2+z^2)
 
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Well, just ask yourself if you recognise the direction the vector field is pointing along at any given coordinate. In particular, consider the position vector of that coordinate as well. Doesn't this look familiar?
 
First obvious point: For any vector field, choose some points (x,y,z) ( because you can't plot all of them!), calculate the vector at that point, and plot it.

Now, look at Defennder's post for this specific vector field. \sqrt{x^2+ y^2+ z^2} is a number and affects the length of the post. How does being closer or farther away from the origin affect the length of the vector? The direction is given entirely by xi+yj+zk. What direction would that be starting at the point (x,y,z)?
 

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