Chapter 1: Vectors, Lines and Planes
Section 1.7: Planes
Example 1.7.8
Obtain an equation for λ, the line that passes through the point A:5,1,−3, intersects L, the line x=2+s,y=3 s−1,z=4−2 s, and is parallel to P, the plane 3 x−2 y+ z+7=0.
Solution
Mathematical Solution
The search for line λ begins with a "diagnosis" in which a strategy for its determination is proposed.
Strategy
As per Figure 1.7.8(a), let Q (pink) be a plane through point A (green) and parallel to plane P (gray).
Let B (black) be the point of intersection of line L (red) with plane Q.
The line through A and B is the line λ.
Figure 1.7.8(a) Planes P and Q, points A and B, and lines L and λ
Implementation
As a first step, plane Q must be found. Since it is parallel to the given plane P, it has the same normal, N. Hence, plane Q is given by R−A·N=0, that is, by
(xyz−51−3)·3−21 = 3x−5−2y−1+z+3=3 x−2 y+z−10=0
The intersection of line L with plane Q is found by solving 3 xs−2 ys+zs−10=0, that is,
32+s−23 s−1+4−2 s−10=0
for s=2/5 so point B is then xs,ys,zsx=a|f(x)s=2/5 = 12/5,1/5,16/5.
Finally, λ, the line through both A and B, is given by R=A+t B−A, that is, by
xyz=51−3+t (12/51/516/5−51−3) = 51−3+t5−13−431
Maple Solution - Interactive
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Define the plane P
Control-drag the equation of plane P.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Plane
Context Panel: Assign to a Name≻P
3 x−2 y+ z+7=0→make plane<< Plane 1 >>→assign to a nameP
Define the line L
Using the technique of control-drag, make a list of the parametric equations for line L.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Line
Context Panel: Assign to a Name≻L
x=2+s,y=3 s−1,z=4−2 s→make line<< Line 1 >>→assign to a nameL
Obtain N, the normal for plane P
Write the name P.
Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Normal
Context Panel: Assign to a Name≻N
P = << Plane 1 >>→normal →assign to a nameN
Obtain plane Q
Form a sequence of point A and the name of plane P.
Context Panel: Assign to a Name≻Q
5,1,3,P→make plane<< Plane 2 >>→assign to a nameQ
Obtain point B, the intersection of line L and plane Q
Write the sequence of names for line L and plane Q.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Intersection
Context Panel: Assign to a Name≻B
L,Q→intersection65,−175,285→assign to a nameB
Obtain line λ
Write a sequence of point A and the name of point B.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation≻vectors
5,1,−3,B→make line<< Line 5 >>→representation
Maple Solution - Coded
Table 1.7.8(a) uses the command-form of the "Lines & Planes" tools in the Student MultivariateCalculus package to obtain an equation for line λ.
Use the Plane command to define plane P.
P≔Plane3 x−2 y+ z+7=0:
Use the Line command to define line L.
L≔Linex=2+s,y=3 s−1,z=4−2 s:
Define point A.
A≔5,1,−3:
Use the GetNormal command to obtain N, the normal on plane P.
N≔GetNormalP:
Use the Plane command to obtain Q, the plane parallel to P and through A.
Q≔PlaneA,N:
Use the GetIntersection command to obtain B, the intersection of line L with plane Q.
B≔GetIntersectionL,Q:
Use the Line command to obtain λ, the line through points A and B.
λ≔LineA,B:
Use the GetRepresentation command to display the vector form of line λ.
GetRepresentationλ,form=vectors
Table 1.7.8(a) Command-based determination of line λ
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