Chapter 1: Vectors, Lines and Planes
Section 1.6: Lines
Example 1.6.1
Obtain an equation for the line that passes through the point P:3,2,1 and that is parallel to the vector V=2 i−3 j+5 k.
Solution
Mathematical Solution
The vector form of the equation of the line is R=P+t V, where R is the generic position vector to any point on the line, P is the position vector to point P, V is the direction vector for the line, and t is the parameter along the line.
R=xyz=321+t 2−35
The value of t is controlled by the slider in Figure 1.6.1(a). As t varies, the head of the position vector R, drawn in red and emanating from the yellow dot that delineates the origin, traverses the line. The direction vector V is in green, and point P is represented by the black dot in the figure.
t = =
Figure 1.6.1(a) The line R=P+t V
The parametric form of the equation of this line is x=3+2 t,y=2−3 t,z=1+5 t.
The symmetric form of the equation of this line is x−32=y−2−3=z−15.
Maple Solution - Interactive
Initialize
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Represent point P as a list. Context Panel: Assign to a Name≻P
3,2,1→assign to a nameP
Enter V as per Table 1.1.1.
Context Panel: Assign to a Name≻V
2,−3,5→assign to a nameV
Define the line and assign it the name L
Write the sequence of point and vector. Press the Enter key.
Context Panel: Student Multivariate Calculus≻ Lines & Planes≻Line (Click OK in the dialog for choosing variables. See Figure 1.6.1(b),)
Context Panel: Assign to a Name≻L
Figure 1.6.1(b) Variables dialog
P,V
→make line
<< Line 1 >>
→assign to a name
L
Obtain each of the five possible representations of the line L
Type L and press the Enter key.
Context Panel: Student Multivariate Calculus≻ Lines & Planes≻Representation≻ Select "parametric" from the drop-down. See Figure 1.6.1(c).)
Figure 1.6.1(c) Representations dialog
→representation
x=3+2t,y=2−3t,z=1+5t
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation≻ Select "combined_vector" from the drop-down. See Figure 1.6.1(c).)
L = << Line 1 >>→representation
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation≻ Select "vectors" from the drop-down. See Figure 1.6.1(c).)
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation≻ Select "symmetric" from the drop-down. See Figure 1.6.1(c).)
L = << Line 1 >>→representationx2−32 = −y3+23 = z5−15
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation≻ Select "equations" from the drop-down. See Figure 1.6.1(c).)
L = << Line 1 >>→representation−52x+z=−132,32x+y=132
The drop-down for representation-type in Figure 1.6.1(c) lists the options in alphabetical order. However, the order adopted in this solution has been altered, with the most useful types listed first, and the least useful, last. The two most useful forms for the equation of a line in space are the parametric form, and its close ally, the "combined vector" form. The form called "vectors" can be converted to the "combined vector" form by an application of the Simplify option in the Context Panel. The symmetric form of the line is of little use in solving problems involving lines and planes, and the form called "equations" is not found in the typical calculus textbook.
Maple Solution - Coded
If the Student MultivariateCalculus package has already been loaded, and the point P and direction vector V already defined, then the following commands will define the line L and obtain each of its five possible representations.
Apply the Line command and assign the line structure to the name L.
L≔LineP,V
<< Line 2 >>
Apply the GetRepresentation command with the form option "parametric".
GetRepresentationL,form=parametric
Apply the GetRepresentation command with the form option "combined_vector".
GetRepresentationL,form=combined_vector
Apply the GetRepresentation command with the form option "vectors".
GetRepresentationL,form=vectors
Apply the GetRepresentation command with the form option "symmetric".
GetRepresentationL,form=symmetric
x2−32 = −y3+23 = z5−15
Apply the GetRepresentation command with the form option "equations".
GetRepresentationL,form=equations
−52x+z=−132,32x+y=132
<< Chapter Overview 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
