Chapter 1: Vectors, Lines and Planes
Section 1.6: Lines
Example 1.6.5
Lines L1 and L2 both have the common direction V=3 i−2 j+5 k, with L1 passing through point P:7,−4,6 and L2 passing through Q:4,−7,1.
Find the equations of L1 and L2.
Calculate the distance between L1 and L2.
Solution
Mathematical Solution
Part (a)
If P and Q are position vectors to points P and Q, respectively, then
line L1 is described vectorially by R=P+t V, that is, by xyz=7−46+t 3−25 and
line L2 is described vectorially by R=Q+s V, that is, by xyz=4−71+s 3−25.
Part (b)
Figure 1.6.5(a) is a sketch of the parallel lines L1 and L2, along with the points P and Q, the direction vector V, the vector U from point Q to point P, and U⊥V, the component of U orthogonal to V.
The distance between the points is the magnitude of U⊥V=U−UV, where UV is the projection of U onto V.
Obtain the vector
U=P−Q=7−46−4−71=335
Figure 1.6.5(a) Parallel Lines L1 and L2, and vectors U=P−Q and V
then calculate
UV=U·VV·VV=2838V=14193−25 and U⊥V=U−UV=335−14193−25=5193175
so that finally, the distance between the lines is
U⊥V = 5199+289+25=5323/19
Maple Solution - Interactive
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Enter V as per Table 1.1.1.
Context Panel: Assign to a Name≻V
3,−2,5→assign to a nameV
Obtain lines L1 and L2
Form a sequence of point P (or Q) and the name V.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Line
Context Panel: Assign to a Name≻L[k], k=1,2
7,−4,6,V→make line<< Line 1 >>→assign to a nameL1
4,−7,1,V→make line<< Line 2 >>→assign to a nameL2
Display a representation of each line
Write the name of the line. Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation≻vectors (Use t on L1 and s on L2 as the parameters along the lines.)
L1 = << Line 1 >>→representation
L2 = << Line 2 >>→representation
Write the sequence of names for the two lines. Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Distance
Context Panel: Combine≻radical
Context Panel: Approximate≻5 (digits)
L1,L2 = << Line 1 >>,<< Line 2 >>→distance5383438= combine 519323→at 5 digits4.7295
The exact distance between the lines can also be expressed as 517/19, or even 5323/19.
The traditional approach to the calculation of the distance between two parallel lines is vector-based, obtaining, for any vector between the two lines, its scalar projection orthogonal to the common direction of the lines. (See Figure 1.6.5(a).) This calculation is give below.
Define the position vectors P and Q
Enter P as per Table 1.1.1.
Context Panel: Assign to a Name≻P
7,−4,6→assign to a nameP
Enter Q as per Table 1.1.1.
Context Panel: Assign to a Name≻Q
4,−7,1→assign to a nameQ
Obtain U, the vector from Q on L2 to P on L1
Write the definition of U.
Context Panel: Assign Name
U=P−Q→assign
Obtain UV, the projection of U upon V
Write the sequence of names U, V. Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Projection
Context Panel: Assign to a Name≻UV
U,V = →projection →assign to a nameUV
Obtain U⊥V=U−UV and its magnitude
Write the difference of U and UV. Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Norm
U−UV = →norm519323
Maple Solution - Coded
Initialize
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define the position vectors P and Q, and the direction vector V.
P,Q,V≔7,−4,6,4,−7,1,3,−2,5:
Obtain Q, the vector from Q to P.
U≔P−Q:
Calculate U⊥V = U−UV
Apply the Norm and DotProduct commands.
NormU−DotProductU,VDotProductV,V V = 519323
<< Previous Example Section 1.6 Next Section >>
© 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
