How Do You Sketch These Vector Fields in the XY-Plane?

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SUMMARY

The discussion focuses on sketching vector fields in the XY-plane, specifically three vector functions: F(r) = 2r, F(r) = -r/||r||³, and F(x,y) = 4xi + xj. Participants emphasize the importance of analyzing vector direction and magnitude, noting that F(r) = 2r represents a 3D vector field. The correct approach involves understanding the notation for r as r = xi + yj + zk and recognizing that in the XY-plane, z = 0 simplifies the calculations. The discussion concludes that sketching involves evaluating vectors at various points to identify symmetries.

PREREQUISITES
  • Understanding of vector fields and their representations
  • Familiarity with 3D coordinate systems and notation
  • Knowledge of vector magnitude and direction
  • Basic skills in sketching mathematical functions
NEXT STEPS
  • Learn how to visualize 3D vector fields in 2D using projection techniques
  • Study the properties of vector functions, particularly in physics applications
  • Explore symmetry in vector fields and its implications for sketching
  • Investigate the mathematical implications of the function F(r) = -r/||r||³
USEFUL FOR

Students in mathematics or physics, educators teaching vector calculus, and anyone interested in visualizing vector fields in the XY-plane.

-EquinoX-
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Homework Statement



I am asked to sketch the following vector field in the xy-plane

(a) F(r) = 2r


(b) F(r) = -r/||r||3


(c) F(x,y) = 4xi + xj

Homework Equations





The Attempt at a Solution



Can someone please give me some hints on how to proceed
 
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hi EquinoX

first I would look at the direction and magitudes of the vectors & see if you can spot any symmetries

a good way to start is to sketch the vector at several points as well
 
yes but for F(r) = 2r this is a 3D vector field, so if I am asked to sketch it in an xy plane, do I just look at the i + j direction? and is 2r basically just 2xi + 2yj + 2zk?

because r is xi + yj + zk?
 
Last edited:
Yeah i think that should do

r = sqrt(x^2 + y^2 + z^2) and in the xy plane z = 0, so this shouldn't affect your r anyway
 
so r = sqrt(x^2)i + sqrt(y^2)j ?
 
sorry bit of confusion over notation
|r| = sqrt(x^2 + y^2 + z^2)

r = xi + yj + zk
 
I am asking 2r?
 

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