Chapter 1: Vectors, Lines and Planes
Section 1.7: Planes
Example 1.7.9
If P is the plane 2 x− y+3 z=5, obtain an equation for the line λ that lies in P, that is perpendicular to L, the line given parametrically by
x=2−3 t,y=1+2 t,z=3+ t
and that is at a distance d=4 from M, the point of intersection of L with P.
Solution
Mathematical Solution
In Figure 1.7.9(a), the blue arrow represents the vector W, the direction of line L.
Point M, the intersection of line L and plane P, is shown in gold.
The red arrow denotes N, the normal to plane P.
The green arrow represents the vector V=W×N that lies in plane P, and is orthogonal to Q, the plane containing both W and N.
The vectors ±4 U^, lie in the intersection of planes P and Q, and have length 4. Their heads are 4 units from point M.
Line λ will pass through the head of either of ±4 U^ and be parallel to V.
use plots, LinearAlgebra, Student:-VectorCalculus in module() local d,X,Y,Z,P,T,MM,W,N,V,U,R,H1,H2,n,v,u,uu,w,p1,p2,p3,p4,p5,p6,p7,p8,p9,p10; d:=4:
X:=2-3*t: Y:=1+2*t: Z:=3+t:
P:=2*x-1*y+3*z-5:
T:=solve(eval(P,[x=X,y=Y,z=Z]),t): MM:=eval(<X,Y,Z>,t=T): W:=map(coeff,<X,Y,Z>,t): N:=<coeff(P,x),coeff(P,y),coeff(P,z)>: V:=LinearAlgebra:-CrossProduct(W,N)/2: U:=LinearAlgebra:-CrossProduct(V,N): R:=d*U/LinearAlgebra:-Norm(U,2): H1:=M+R+s*V: H2:=M-R+s*V:
n:=RootedVector(root=MM,N): v:=RootedVector(root=MM,V): u:=RootedVector(root=MM,R): uu:=RootedVector(root=MM,-R): w:=RootedVector(root=MM,W):
p1:=PlotVector(n,color=red,width=.2): p2:=PlotVector(v,color=green,width=.2): p3:=PlotVector(u,color=black,width=.2): p4:=PlotVector(uu,color=black,width=.2): p5:=PlotVector(w,color=blue,width=.2): p6:=implicitplot3d(P,x=-5..5,y=1..12,z=0..10,style=surface,transparency=.9,color=red): p7:=textplot3d({[-5.5,6,5,'W'],[2,9.5,3.5,'V'],[.3,3,2,typeset(4*`#mover(mi("U",fontstyle = "normal",fontweight = "bold"),mo("ˆ"))`)],[-.2,2.8,7.9,'N']},font=[Times,Bold,14]): p8:=textplot3d([-11/5, 2.8, 22/5,M],font=[default,14]): p9:=pointplot3d([-2.2,3.8,4.4],symbol=solidsphere,symbolsize=20,color=gold): p10:=display([p1,p2,p3,p4,p5,p6,p7,p8,p9],axes=none,scaling=constrained,lightmodel=none,orientation=[90,75,85]): print(p10); end module: end use:
Figure 1.7.9(a) Normal (red), direction of L (blue), direction of λ (green), point M (gold),vectors orthogonal to λ and of length 4 (black)
Table 1.7.9(a) contains a schematic for a vector-based determination of line λ.
L intersects P at M:−11/5,19/5,22/5.
W= −3 i+2 j+k is direction of L.
N=2 i−j+3 k is normal to P.
V=W×N is the direction of λ.
U=V×N is orthogonal to V and lies in the intersection of planes P and Q.
U^=U/U
λ, with direction V, is at the tip of ±4 U^
Table 1.7.9(a) Schematic for vector-based determination of line λ
Maple Solution - Interactive
Table 1.7.9(b) lists the steps by means of which the requisite lines can be found with the "Lines & Planes" tools in the Student MultivariateCalculus package.
Define plane P and line L.
Obtain point M as the intersection of line L and plane P.
Obtain N, the normal for plane P.
Obtain W, the direction vector along line L.
Obtain Q, the plane containing point M and the directions N and W.
Obtain V, the normal for plane Q.
Obtain the line of intersection of planes Q and P, then extract U, the direction of this line. Normalize U, calling the unit vector so formed u instead of U^.
Obtain m, the position vector to point M.
Form m ±4 u+sV, the vector-form of the lines in plane P, 4 units from point M.
Table 1.7.9(b) Steps for finding the requisite lines with the "Lines & Planes" tools in the Student MultivariateCalculus package
Table 1.7.9(c) implements the calculations listed in Table 1.7.9(b).
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Define plane P
Control-drag the equation of plane P.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Plane
Context Panel: Assign to a Name≻P
2 x− y+3 z=5→make plane<< Plane 1 >>→assign to a nameP
Define line L
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=2−3 t,y=1+2 t,z=3+ t→make line<< Line 1 >>→assign to a nameL
Obtain point M, the intersection of line L with plane P
Form a sequence of the names for the line L and plane P.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Intersection
Context Panel: Assign to a Name≻M
L,P→intersection−115,195,225→assign to a nameM
Obtain the normal for plane P
Write the name of plane P.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Normal
Context Panel: Assign to a Name≻N
P→normal2−13→assign to a nameN
Obtain W, the direction vector along line L
Write the name of line L.
Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Direction
Context Panel: Assign to a Name≻W
L→direction−321→assign to a nameW
Obtain plane Q that contains point M and the directions N and W
Write the sequence of names for point M, and vectors N and W.
Context Panel: Assign to a Name≻Q
M,N,W→make plane<< Plane 2 >>→assign to a nameQ
Obtain V, the normal for plane Q
Write the name of plane Q.
Context Panel: Assign to a Name≻V
Q→normal−7−111→assign to a nameV
Obtain u, a unit vector in the direction of the line of intersection of planes Q and P
Write the sequence of names for planes Q and P.
Context Panel: Student Multivariate Calculus≻Normalize
Context Panel: Assign to a Name≻u
Q,P→intersection<< Line 2 >>→direction−322329→normalize−162663992326679829266798→assign to a nameu
Obtain m, the position vector to point M
Write the name of point M.
Context Panel: Conversions≻Column Vector
M = −115,195,225→to Vector−115195225→assign to a namem
Obtain the equations of the lines that pass through the heads of m ±4 u and that have direction V
m+4 u+s V
−115−64266399−7s195+46266399−11s225+58266399+s
m−4 u+s V
−115+64266399−7s195−46266399−11s225−58266399+s
Table 1.7.9(c) Implementation of the calculations listed in Table 1.7.9(b)
Maple Solution - Coded
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define plane P via the Plane command.
P≔Plane2 x− y+3 z=5:
Define line L via the Line command.
L≔Linex=2−3 t,y=1+2 t,z=3+ t:
Obtain N, the normal on plane P via the GetNormal command.
N≔GetNormalP:
Obtain point M, the intersection of line L with plane P, via the GetIntersection command.
M≔GetIntersectionL,P:
Obtain W, the direction of line L, via the GetDirection command.
W≔GetDirectionL:
Obtain Q, the plane containing point M, and the directions W and N, via the Plane command.
Q≔PlaneM,W,N:
Obtain V, the normal on plane Q, via the GetNormal command.
V≔GetNormalQ:
Get u, a unit vector along the line of intersection of planes Q and P by using the GetIntersection, GetDirection and Normalize commands.
u≔NormalizeGetDirectionGetIntersectionQ,P:
m≔VectorM:
−115+64266399+7s195−46266399+11s225−58266399−s
−115−64266399+7s195+46266399+11s225+58266399−s
A vector-based calculation that eschews the power of the Student MultivariateCalculus package will require a greater knowledge of Maple syntax and more human intervention, as the calculations in Table 1.7.9(d) show.
Intersect line L with plane P and extract the intersection point as the position vector m
T≔solveeval2 x− y+3 z=5,x=2−3 t,y=1+2 t,z=3+ t:M≔evalx=2−3 t,y=1+2 t,z=3+ t,t=T:m≔evalx,y,z,M:
By inspection, write the vectors N and W
N,W≔2,−1,3,−3,2,1:
Obtain V=W×N and u=V×NV×N
V≔CrossProductW,N:
u≔NormalizeCrossProductV,N:
Obtain the two lines m ±4 u+s V
Table 1.7.9(d) Vector-based calculation of the requisite lines.
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