Chapter 1: Vectors, Lines and Planes
Section 1.7: Planes
Example 1.7.5
Find an equation for S3, the plane that contains the point P:2,−3,1 and the line of intersection of the planes
S1: 3 x−7 y−9 z=8 and S2:5 x+4 y−2 z=6
Solution
Mathematical Solution
Figure 1.7.5(a) shows the given planes S1 and S2 in red, and the solution plane S3 in green.
The red arrow represents N3, the normal to plane S3.
The black line is the line of intersection of S1 and S2, while the gold arrow represents V, the direction vector for the line of intersection.
The black arrow represents the vector W=P−A, where A is an arbitrary point on the line of intersection.
The gold "dot" at the head of the black arrow marks the location of point P.
use plots, VectorCalculus in module() local p1,p2,p3,p4,p5,p6,M,H,N3: M:=RootedVector(root=[74/47, -22/47, 0],<2,-3,1>-<74/47, -22/47, 0>): H:=RootedVector(root=[74/47, -22/47, 0],<50/47, -39/47, 1>): N3:=RootedVector(root=[74/47, -22/47, 0],<-80/47, 30/47, 110/47>): p1:=implicitplot3d([3*x-7*y-9*z = 8,5*x+4*y-2*z = 6],x=-5..5,y=-5..5,z=-5..5,transparency=.9,style=surface,color=red): p2:=spacecurve([74/47+(50/47)*t, -22/47-(39/47)*t, t],t=-3..3,color=black,thickness=5,numpoints=2): p3:=implicitplot3d([14-8*x+3*y+11*z = 0],x=-5..5,y=-5..5,z=-5..5,style=surface,color=green): p4:=PlotVector([M,H,N3],color=[black,gold,red],width=.3): p5:=pointplot3d([2,-3,1],symbol=solidsphere,symbolsize=25,color=gold): p6:=display(p1,p2,p3,p4,p5,scaling=constrained,axes=none,view=[0..5,-4..2,-2..5],lightmodel=none,orientation=[20,40,-20]): print(p6); end module: end use:
Figure 1.7.5(a) Planes S1, S2 (red), and S3 (green), vectors V (gold), W (black) and N3 (red)
Figure 1.7.5(a), which summarizes the path of the calculation, can be rotated with the mouse. The calculations themselves are as follows.
If N1 and N2 are respectively the normals to planes S2 and S2, then
V=N1×N2 = |ijk3−7−954−2| = 50−3947
An arbitrary point on the line of intersection can be found by setting z=0 in the equations for planes S1 and S2, and solving for A = 74/47,−22/47,0; hence,
W=P−A = 2−31−74/47−22/470 = 20/7−119/471
The normal to plane S3 is then
N3=V×W = |ijk50−394720/47−119/471| = 80−30−110
Implementing R−A·N3=0 as the equation of the plane S3 leads to
(xyz−74/47−22/470)·80−30−110 = 80x−74/47−30y+22/47−110 z=0
which simplify to 8 x−3 y−11 z=14.
Maple Solution - Interactive
The tools for defining and manipulating lines and planes in the Student MultivariateCalculus package allow for a much shortened calculation. In effect, once the planes S1 and S2 are defined, L, the line of their intersection becomes available. Then, plane S3 is immediately defined by point P and line L.
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Define planes S1 and S2
Control-drag the equation for plane S1
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Plane
Context Panel: Assign to a Name≻S[k] (k=1,2)
3 x−7 y−9 z=8→make plane<< Plane 1 >>→assign to a nameS1
5 x+4 y−2 z=6→make plane<< Plane 2 >>→assign to a nameS2
Obtain L, the line of intersection of planes S1 and S2
Write the sequence of names for planes S1 and S2
Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Line
Context Panel: Assign to a Name≻L
S1,S2 = << Plane 1 >>,<< Plane 2 >>→make line<< Line 1 >>→assign to a nameL
Obtain the equation of plane S3 that contains line L and point P:2,−3,1
Write the sequence consisting of point P (as a list) and the name L.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation
2,−3,1,L→make plane<< Plane 3 >>→representation8x−3y−11z=14
The traditional, but more tedious vector-based solution, is implemented as follows.
Define N1 and N2, the normals to planes S1 and S2, respectively
Context Panel: Assign to a Name≻N[1]
3,−7,−9→assign to a nameN1
Context Panel: Assign to a Name≻N[2]
5,4,−2→assign to a nameN2
Obtain V, the direction of the line of intersection
Common Symbols palette: Cross-product operator
Context Panel: Assign Name
V=N1×N2→assign
Obtain (as the position vector A), a point on the line of intersection
Form a sequence of equations for planes S1, S2 Press the Enter key.
Context Panel: Evaluate at a Point≻z=0
Context Panel: Solve≻Solve
Context Panel: Assign to a Name≻s
Expression palette: Evaluation template Evaluate x,y,0 at the solution s
Context Panel: Assign to a Name≻A
3 x−7 y−9 z=8,5 x+4 y−2 z=6
3x−7y−9z=8,5x+4y−2z=6
→evaluate at point
3x−7y=8,5x+4y=6
→solve
x=7447,y=−2247
→assign to a name
s
x,y,0x=a|f(x)s = →assign to a nameA
Define the position vectors P and R
Context Panel: Assign to a Name≻P
2,−3,1→assign to a nameP
Context Panel: Assign to a Name≻R
x,y,z→assign to a nameR
Obtain W=P−A, a second vector in plane S3
W=P−A→assign
Obtain N3=V×W, the normal to plane S3
Common Symbols palette: Cross product operator
N3=V×W→assign
Implement the vector form of the equation of a plane
Common Symbols palette: Dot product operator Press the Enter key.
R−A·N3=0
80x−30y−110z−140=0
Elementary algebraic manipulation leads finally to 8 x−3 y−11 z=14.
Maple Solution - Coded
Using the "lines and planes" tools in the Student MultivariateCalculus package, the equation of plane S3 would be obtained as per the calculations in Table 1.7.5(a).
Use the Plane command to define plane S1.
S1≔Plane3 x−7 y−9 z=8:
Use the Plane command to define plane S2.
S2≔Plane5 x+4 y−2 z=6:
Use the Line command to define L, the line of intersection of planes S1 and S2.
L≔LineS1,S2:
Apply the Plane command to point P and line L to obtain plane S3.
S3≔Plane2,−3,1,L:
Apply the GetRepresentation command to S3 to obtain the equation of this plane.
GetRepresentationS3
8x−3y−11z=14
Table 1.7.5(a) The equation of plane S3 via the "lines and planes" tools
The traditional vector-based calculation can build on the results in Table 1.7.5(a).
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Obtain V=N1×N2
Obtain normals to planes S1 and S2 via the GetNormal command; then apply the CrossProduct command
V≔CrossProductGetNormalS1,GetNormalS2 =
Obtain W=P−A^, where A^ is a random point on line L
Apply the GetPoint command to line L to obtain an arbitrary point on this line. Use the Vector command to convert the list to a vector.
W≔P− VectorGetPointL
Obtain N3=V×W as the normal to plane S3
Apply the CrossProduct command.
N3≔CrossProductV,W
Implement the vector form of the plane S3 as N3·R−P=0
Apply the DotProduct command.
DotProductN3,R−P=0
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