Example 1-7-12 - Maple Help

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Home : Support : Online Help : Education : Study Guides : MultivariateCalculus : Chapter 1 : Examples : Section 1-7 : Example 1-7-12

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 $P_{1}$ , the plane $3 x + y + 2 z = 28$ , and makes an angle of $π / 4$ with $P_{2}$ , the plane $4 x + 3 y - z = 1$ .

Solution

Mathematical Solution

•	There are two possible planes $Q_{1}$ and $Q_{2}$ satisfying the conditions of the example, as illustrated in Figure 1.7.12(a).

•	The given planes $P_{1}$ and $P_{2}$ are drawn in gray and gold, respectively; point M is the black dot; and the red and green planes are $Q_{1}$ and $Q_{2}$ .

•	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 $N \cdot M = d$ , where M is the position vector to M.

•	If U is the normal for $P_{1}$ , then $Q$ is perpendicular to $P_{1}$ if $N \cdot U = 0$ .

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 $P_{1}$ (gray), $P_{2}$ (gold); point M (black); solution planes $Q_{1}$ (red), $Q_{2}$ (green)

•	Planes $Q$ and $P_{2}$ meet at an angle of $π / 4$ if $\cos (π / 4) = \frac{1}{\sqrt{2}}$ = $\frac{N \cdot V}{∥N ∥∥ V∥}$ , where $V$ is a normal for plane $P_{2}$ .

•	These three conditions are expressed by the equations

$2 a - b + c = d, 3 a + b + 2 c = 0, 4 a + 3 b - c = \frac{1}{2} \sqrt{a^{2} + b^{2} + c^{2}} \sqrt{26} \sqrt{2}$

•	The solutions of these equations are

$\{a = - \frac{2}{11} d, b = - \frac{8}{11} d, c = \frac{7}{11} d\}$ and $\{a = - \frac{8}{5} d, b = - \frac{6}{5} d, c = 3 d\}$

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

$⟨a, b, c⟩$ $\overset{assign to a name}{\to}$ $N$

•	Enter U as per Table 1.1.1.

•	Context Panel: Assign to a Name≻U

$⟨3, 1, 2⟩$ $\overset{assign to a name}{\to}$ $U$

•	Enter V as per Table 1.1.1.

•	Context Panel: Assign to a Name≻V

$⟨4, 3, - 1⟩$ $\overset{assign to a name}{\to}$ $V$

Define the position vectors M and R

•	Enter M as per Table 1.1.1.

•	Context Panel: Assign to a Name≻M

$⟨2, - 1, 1⟩$ $\overset{assign to a name}{\to}$ $M$

•	Enter R as per Table 1.1.1.

•	Context Panel: Assign to a Name≻R

$⟨x, y, z⟩$ $\overset{assign to a name}{\to}$ $R$

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≻ $S$

$N \cdot M = d, N \cdot U = 0, N \cdot V = ∥N∥ \cdot &par; V &par; \cdot \cos (π / 4)$

$2 a - b + c = d, 3 a + b + 2 c = 0, 4 a + 3 b - c = \frac{1}{2} \sqrt{a^{2} + b^{2} + c^{2}} \sqrt{26} \sqrt{2}$

$\overset{solve (specified)}{\to}$

$\{a = - \frac{2}{11} d, b = - \frac{8}{11} d, c = \frac{7}{11} d\}, \{a = - \frac{8}{5} d, b = - \frac{6}{5} d, c = 3 d\}$

$\overset{assign to a name}{\to}$

$S$

Evaluate $(R - M) \cdot N / d = 0$ , the vector form of a plane, for each solution in $S$

•	Expression palette: Evaluation template Evaluate $(R - M) \cdot N / d = 0$ for each solution in the list $S$

•	Context Panel: Simplify≻Simplify

$\binom{(R - M) \cdot N / d = 0}{} | \binom{}{S_{1}}$

$\frac{- \frac{2}{11} (x - 2) d - \frac{8}{11} (y + 1) d + \frac{7}{11} (z - 1) d}{d} = 0$

$\overset{simplify}{=}$

$- \frac{2}{11} x - 1 - \frac{8}{11} y + \frac{7}{11} z = 0$

$\binom{(R - M) \cdot N / d = 0}{} | \binom{}{S_{2}}$

$\frac{- \frac{8}{5} (x - 2) d - \frac{6}{5} (y + 1) d + 3 (z - 1) d}{d} = 0$

$\overset{simplify}{=}$

$- \frac{8}{5} x - 1 - \frac{6}{5} y + 3 z = 0$

Maple Solution - Coded

•	Install the Student MultivariateCalculus package.

$with (Student :- MultivariateCalculus) &colon;$

Define the vectors N, U, V, M, and R

$N, U, V, M, R ≔ ⟨a, b, c⟩, ⟨3, 1, 2⟩, ⟨4, 3, - 1⟩, ⟨2, - 1, 1⟩, ⟨x, y, z⟩ &colon;$

Solve the three equations $N \cdot M = d, N \cdot U = 0, N \cdot V = ∥N∥ \cdot ∥ V ∥ \cdot \cos (π / 4)$

•	Apply the DotProduct, Angle, and solve commands.

$S ≔ solve (\{DotProduct (M, N) = d, DotProduct (U, N) = 0, Angle (V, N) = π / 4\}, \{a, b, c\}) &colon;$

Evaluate $(R - M) \cdot N / d = 0$ , the general equation of a plane through M, for each solution in $S$

•	Apply the DotProduct, eval, and simplify commands.

$simplify (eval (DotProduct (N, R - M) / d = 0, S_{1}))$

$- \frac{2}{11} x - 1 - \frac{8}{11} y + \frac{7}{11} z = 0$

$simplify (eval (DotProduct (N, R - M) / d = 0, S_{2}))$

$- \frac{8}{5} x - 1 - \frac{6}{5} y + 3 z = 0$

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