Chapter 1: Vectors, Lines and Planes
Section 1.7: Planes
Example 1.7.12
Obtain an equation for Q, the plane that passes through the point M:2,−1,1, is perpendicular to P1, the plane 3 x+ y+2 z=28, and makes an angle of π/4 with P2, the plane 4 x+3 y− z=1.
Solution
Mathematical Solution
There are two possible planes Q1 and Q2 satisfying the conditions of the example, as illustrated in Figure 1.7.12(a).
The given planes P1 and P2 are drawn in gray and gold, respectively; point M is the black dot; and the red and green planes are Q1 and Q2.
The general equation for a plane is
a x+b y+c z=d
If N is a normal for Q, the condition that Q contains the point M is N·M=d, where M is the position vector to M.
If U is the normal for P1, then Q is perpendicular to P1 if N·U=0.
use plots,Student:-LinearAlgebra in module() local N,N1,N2,M,R,S,P,p1,p2,q1,q2,q3,Pa,Pb,g1,g2,G; N:=<a,b,c>: N1:=<3,1,2>: N2:=<4,3,-1>: M:=<2,-1,1>: R:=<x,y,z>: p1:=(R-<3,5,7>).N1: p2:=(R-<2,-1,4>).N2: q1:=(N.M)=d: q2:=N.N1=0: q3:=simplify(VectorAngle(N,N2)=Pi/4): S:=solve({q1,q2,q3},{a,b,c},Explicit): P:=a*x+b*y+c*z=d: Pa:=simplify(eval(P,S[1])/d): Pb:=simplify(eval(P,S[2])/d): g1:=implicitplot3d([Pa,Pb,p1,p2],x=-3..10,y=-5..5,z=-3..10,color=[red,green,gray,gold]): g2:=pointplot3d(M,symbol=solidsphere,symbolsize=30,color=black): G:=display(g1,g2,style=surface,axes=none,scaling=constrained,orientation=[-82,-82,23],lightmodel=none): print(G); end module: end use:
Figure 1.7.12(a) Given planes P1 (gray), P2 (gold); point M (black); solution planes Q1 (red), Q2 (green)
Planes Q and P2 meet at an angle of π/4 if cosπ/4=12 = N·VN V, where V is a normal for plane P2.
These three conditions are expressed by the equations
2a−b+c=d,3a+b+2c=0,4a+3b−c=12a2+b2+c2262
The solutions of these equations are
a=−211d,b=−811d,c=711d and a=−85d,b=−65d,c=3d
so the two candidates for plane Q are defined by the equations
2 x+8 y−7 z= −11 and 8 x+6 y−15 z=−5
Maple Solution - Interactive
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Define the vectors N, U, and V
Enter N as per Table 1.1.1.
Context Panel: Assign to a Name≻N
a,b,c→assign to a nameN
Enter U as per Table 1.1.1.
Context Panel: Assign to a Name≻U
3,1,2→assign to a nameU
Enter V as per Table 1.1.1.
Context Panel: Assign to a Name≻V
4,3,−1→assign to a nameV
Define the position vectors M and R
Enter M as per Table 1.1.1.
Context Panel: Assign to a Name≻M
2,−1,1→assign to a nameM
Enter R as per Table 1.1.1.
Context Panel: Assign to a Name≻R
x,y,z→assign to a nameR
Write and solve the three equations that reflect the conditions imposed on plane Q
Write the sequence of equations and press the Enter key.
Context Panel: Solve≻Solve for Variables≻a,b,c
Context Panel: Assign to a Name≻S
N·M=d,N·U=0,N·V = N⋅∥V∥⋅cosπ/4
→solve (specified)
a=−211d,b=−811d,c=711d,a=−85d,b=−65d,c=3d
→assign to a name
S
Evaluate R−M·N/d=0, the vector form of a plane, for each solution in S
Expression palette: Evaluation template Evaluate R−M·N/d=0 for each solution in the list S
Context Panel: Simplify≻Simplify
R−M·N/d=0x=a|f(x)S1
−211x−2d−811y+1d+711z−1dd=0
= simplify
−211x−1−811y+711z=0
R−M·N/d=0x=a|f(x)S2
−85x−2d−65y+1d+3z−1dd=0
−85x−1−65y+3z=0
Maple Solution - Coded
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define the vectors N, U, V, M, and R
N,U,V,M,R≔a,b,c,3,1,2,4,3,−1,2,−1,1,x,y,z:
Solve the three equations N·M=d,N·U=0,N·V = N⋅∥V∥⋅cosπ/4
Apply the DotProduct, Angle, and solve commands.
S≔solveDotProductM,N=d,DotProductU,N=0,AngleV,N=π/4,a,b,c:
Evaluate R−M·N/d=0, the general equation of a plane through M, for each solution in S
Apply the DotProduct, eval, and simplify commands.
simplifyevalDotProductN,R−M/d=0,S1
simplifyevalDotProductN,R−M/d=0,S2
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