Chapter 1: Vectors, Lines and Planes
Section 1.6: Lines
Example 1.6.2
Obtain an equation for the line through the points P:3,2,1 and Q:5,−1,4.
Solution
Mathematical Solution
Let P and Q be the position vectors to points P and Q, and let R be the position vector to the generic point x,y,z. Then a parametric representation of the line through P and Q is given by
R=P+t Q−P
where t is the parameter along the line. At t=0, R=P; at t=1, R=Q. For the given points, the line is
R=xyz=321+t5−14−321=3+2 t2−3 t1+3 t
Maple Solution - Interactive
Initialize
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Obtain the line through P and Q and assign it the name L
Write a sequence of two points.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Line
Context Panel: Assign to a Name≻L
3,2,1,5,−1,4→make line<< Line 1 >>→assign to a nameL
Get the two most useful representations of the line L
Write the name L. Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation≻parametric (See Figure 1.6.2(a).)
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation≻combined_vector
L = << Line 1 >>→representationx=3+2t,y=2−3t,z=1+3t
L = << Line 1 >>→representation
Figure 1.6.2(a) Representation dialog
The alternative vector-based "traditional" solution appears below.
Define the position vectors P and Q
Enter P as per Table 1.1.1.
Context Panel: Assign to a Name≻P
3,2,1→assign to a nameP
Enter Q as per Table 1.1.1.
Context Panel: Assign to a Name≻Q
5,−1,4→assign to a nameQ
Obtain V, a vector from point P to point Q
Context Panel: Assign Name
V=Q−P→assign
Obtain the "combined vectors" form of the line
Write the vector form R=P+t V Press the Enter key.
x,y,z=P+t V
Obtain the "parametric" form of the line
Write the sequence of vectors R, P+t V Press the Enter key.
Context Panel: Equate
x,y,z,P+t V
→equate
x=3+2t,y=2−3t,z=1+3t
Maple Solution - Coded
If the Student MultivariateCalculus package has been loaded, then the following commands will define the line and provide its two most useful representations.
Apply the Line command, assigning the line to the name L. Press the Enter key.
L≔Line3,2,1,5,−1,4
<< Line 3 >>
Apply the GetRepresentation command. Press the Enter key.
GetRepresentationL,form=parametric
GetRepresentationL,form=combined_vector
<< Previous Example Section 1.6 Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
