Student[MultivariateCalculus] Examples - Maple Help

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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

$\mathrm{with}(\mathrm{Student}:-\mathrm{MultivariateCalculus}):$



Example 1: Equation of a Plane



 Obtain the equation of the plane containing the three points $\left(1,2,3\right)$, $\left(-1,3,1\right)$, $\left(2,1,-1\right)$.



 • 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

$\left[1,2,3\right],\left[-1,3,1\right],\left[2,1,-1\right]$$\stackrel{\text{make plane}}{\to }$${\mathrm{Student}}{:-}{\mathrm{MultivariateCalculus}}{:-}{\mathrm{Plane}}\left(\left[\begin{array}{r}-6\\ -10\\ 1\end{array}\right]{,}\left[{1}{,}{2}{,}{3}\right]{,}{\mathrm{variables}}{=}\left[{x}{,}{y}{,}{z}\right]{,}{\mathrm{id}}{=}{1}\right)$$\stackrel{\text{representation}}{\to }$${-}{6}{x}{-}{10}{y}{+}{z}{=}{-23}$



Example 2: Skew Lines



 Show that  and  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≻$\mathrm{L1}$ (or $\mathrm{L2}$, as appropriate)

$\stackrel{\text{make line}}{\to }$${\mathrm{Student}}{:-}{\mathrm{MultivariateCalculus}}{:-}{\mathrm{Line}}\left(\left[{1}{,}{2}{,}{3}\right]{,}\left[\begin{array}{r}2\\ -3\\ 5\end{array}\right]{,}{\mathrm{variables}}{=}\left[{x}{,}{y}{,}{z}\right]{,}{\mathrm{parameter}}{=}{t}{,}{\mathrm{id}}{=}{1}\right)$$\stackrel{\text{assign to a name}}{\to }$${\mathrm{L1}}$

$\stackrel{\text{make line}}{\to }$${\mathrm{Student}}{:-}{\mathrm{MultivariateCalculus}}{:-}{\mathrm{Line}}\left(\left[{3}{,}{8}{,}{7}\right]{,}\left[\begin{array}{r}-1\\ 0\\ 6\end{array}\right]{,}{\mathrm{variables}}{=}\left[{x}{,}{y}{,}{z}\right]{,}{\mathrm{parameter}}{=}{s}{,}{\mathrm{id}}{=}{2}\right)$$\stackrel{\text{assign to a name}}{\to }$${\mathrm{L2}}$

Verify the lines are skew

 • Context Panel: Student Multivariate Calculus≻Lines & Planes≻Skew (or Parallel or Intersects)

 $\mathrm{L1},\mathrm{L2}$$\stackrel{\text{skew lines?}}{\to }$${\mathrm{true}}$ $\mathrm{L1},\mathrm{L2}$$\stackrel{\text{parallel?}}{\to }$${\mathrm{false}}$ $\mathrm{L1},\mathrm{L2}$$\stackrel{\text{intersect?}}{\to }$${\mathrm{false}}$

Obtain the distance between the lines

 • Context Panel: Student Multivariate Calculus≻Lines & Planes≻Distance
 • Context Panel: Approximate≻10 (digits)

$\mathrm{L1},\mathrm{L2}$$\stackrel{\text{distance}}{\to }$$\frac{{75}\sqrt{{622}}}{{311}}$$\stackrel{\text{at 10 digits}}{\to }$${6.014452050}$



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)

 $\mathrm{L1}$$\stackrel{\text{direction}}{\to }$$\left[\begin{array}{r}2\\ -3\\ 5\end{array}\right]$