notes / updates | multivariable calculus

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November 3, 2025 | Plotting Vector-Valued Functions

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October 15, 2025 | Problem 57, Section 11.5 with SageMath

    
      # Section 11.4 | Problem 57

t = var('t')
P = sphere((0,2,1), 0.2, color='red')
Q = sphere((0,1,2), 0.2, color='purple')
line1 = parametric_plot3d((2*t,1-t,2+t),(t,-3,3))

v = vector([2,-1,1])
u = vector([0,1,-1])
proj = ((u.dot_product(v))/(norm(v)*norm(v)))*v
norml = u - proj
line2 = parametric_plot3d((0 +norml[0]*t,2+norml[1]*t,1+norml[2]*t),(t,-3,3))
P + Q + line1 + line2
    
  

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September 25, 2024 | Some code for SageMath and Geogebra

  
  # SageMath start
  # To declare Point objects
  A=(1,2,3)
  B=(1,-2,5)
  C=(3,1,5)

  # To draw a vector as a geometric shape
  arrow(A,B)
  # To draw more than 1 vector (triangle rule)
  arrow(A,B) + arrow(B,C) + arrow (A,C)

  # To form a vector from points
  AB = vector(B[0]-A[0], B[1]-A[1], B[2]-A[2])
  BC = vector(C[0]-B[0], C[1]-B[1], C[2]-B[2])

# To print the vector:
 AB 
 print("BC: ", BC)

 # To plot vectors
 plot(AC, start=A) + plot(AB, start=A) + plot(BC, start=B)

 # To check if two vectors are equal
  
 AB == BC
   
  # SageMath end

   # Geogebra start (save the document if you want to keep your work)
   #declare Point objects
   A=Point({1,1,1})
   B=Point({-1,2,3})
   C=Point({2,2,-1})

   # form vectors from points
   AB = Vector(A,B)
   BC = Vector(B,C)

   # Geogebra end
  
  

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