Student MultivariateCalculus Examples
The Student:-MultivariateCalculus package is designed to aid in the teaching and understanding of multivariate calculus concepts. For a general overview, see MultivariateCalculus. For introductory examples, see MultivariateCalculus Example Worksheet.
Lines and Planes
Initialization
Tools≻Load Package: Student Multivariate Calculus
with(Student:-MultivariateCalculus):
Example 1: Equation of a Plane
Obtain the equation of the plane containing the three points 1,2,3, −1,3,1, 2,1,−1.
Write a sequence of the three points.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Plane In the "Choose Variables for Plane" dialog, accept default names or provide new ones.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation
1,2,3,−1,3,1,2,1,−1→make planeStudent:-MultivariateCalculus:-Plane⁡−6−101,1,2,3,variables=x,y,z,id=1→representation−6⁢x−10⁢y+z=−23
Example 2: Skew Lines
Show that x=1+2 t,y=2−3 t,z=3+5 t and x=3−s,y=5+3,z=7+6 s define skew lines, and find the distance between them.
Create Line Objects for each line
Form a list of the parametric equations defining a line.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Line≻t or s, as appropriate
Context Panel: Assign to a Name≻L1 (or L2, as appropriate)
x=1+2 t,y=2−3 t,z=3+5 t→make lineStudent:-MultivariateCalculus:-Line⁡1,2,3,2−35,variables=x,y,z,parameter=t,id=1→assign to a nameL1
x=3−s,y=5+3,z=7+6 s→make lineStudent:-MultivariateCalculus:-Line⁡3,8,7,−106,variables=x,y,z,parameter=s,id=2→assign to a nameL2
Verify the lines are skew
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Skew (or Parallel or Intersects)
L1,L2→skew lines?true
L1,L2→parallel?false
L1,L2→intersect?false
Obtain the distance between the lines
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Distance
Context Panel: Approximate≻10 (digits)
L1,L2→distance75⁢622311→at 10 digits6.014452050
The standard approach to finding the distance between skew lines is vectorial: Obtain N, the vector orthogonal to both lines, and project V, any vector from one line to the other, onto N. The length of this projection is the distance between the lines.
Obtain N, the common normal
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Direction
Context Panel: Assign to a Name≻V1 (or V2, as applicable)
L1→direction2−35→assign to a nameV1