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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 $V = W \times N$ that lies in plane $P$ , and is orthogonal to $Q$ , the plane containing both W and N.

•	The vectors $\pm 4 \overset{&Hat;}{U}$ , 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 $\pm 4 \overset{&Hat;}{U}$ 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:=2-3t: Y:=1+2t: Z:=3+t:

>	P:=2x-1y+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:=dU/LinearAlgebra:-Norm(U,2): H1:=M+R+sV: 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("&circ;"))`)],[-.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 $P$ .

•	$V = W \times N$ is the direction of $λ$ .

•	$U = V \times N$ is orthogonal to V and lies in the intersection of planes P and $Q$ .

•	$\overset{&Hat;}{U} = U / ∥U∥$

•	$λ$ , with direction V, is at the tip of $\pm 4 \overset{&Hat;}{U}$

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 $\overset{&Hat;}{U}$ .

•	Obtain m, the position vector to point M.

•	Form $(m \pm 4 u) + s V$ , 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≻ $P$

$2 x - y + 3 z = 5$ $\overset{make plane}{\to}$ $<< Plane 1 >>$ $\overset{assign to a name}{\to}$ $P$

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≻ $L$

$[x = 2 - 3 t, y = 1 + 2 t, z = 3 + t]$ $\overset{make line}{\to}$ $<< Line 1 >>$ $\overset{assign to a name}{\to}$ $L$

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

$L, P$ $\overset{intersection}{\to}$ $[- \frac{11}{5}, \frac{19}{5}, \frac{22}{5}]$ $\overset{assign to a name}{\to}$ $M$

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

$P$ $\overset{normal}{\to}$ $[\begin{array}{c} 2 \\ −1 \\ 3 \end{array}]$ $\overset{assign to a name}{\to}$ $N$

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

$L$ $\overset{direction}{\to}$ $[\begin{array}{c} −3 \\ 2 \\ 1 \end{array}]$ $\overset{assign to a name}{\to}$ $W$

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: Student Multivariate Calculus≻Lines & Planes≻Plane

•	Context Panel: Assign to a Name≻ $Q$

$M, N, W$ $\overset{make plane}{\to}$ $<< Plane 2 >>$ $\overset{assign to a name}{\to}$ $Q$

Obtain V, the normal for plane $Q$

•	Write the name of plane $Q$ .

•	Context Panel: Evaluate and Display Inline

•	Context Panel: Student Multivariate Calculus≻Lines & Planes≻Normal

•	Context Panel: Assign to a Name≻V

$Q$ $\overset{normal}{\to}$ $[\begin{array}{c} −7 \\ −11 \\ 1 \end{array}]$ $\overset{assign to a name}{\to}$ $V$

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≻Lines & Planes≻Intersection

•	Context Panel: Student Multivariate Calculus≻Lines & Planes≻Direction

•	Context Panel: Student Multivariate Calculus≻Normalize

•	Context Panel: Assign to a Name≻u

$Q, P$ $\overset{intersection}{\to}$ $<< Line 2 >>$ $\overset{direction}{\to}$ $[\begin{array}{c} −32 \\ 23 \\ 29 \end{array}]$ $\overset{normalize}{\to}$ $[\begin{array}{c} - \frac{16 \sqrt{266}}{399} \\ \frac{23 \sqrt{266}}{798} \\ \frac{29 \sqrt{266}}{798} \end{array}]$ $\overset{assign to a name}{\to}$ $u$

Obtain m, the position vector to point M

•	Write the name of point M.

•	Context Panel: Evaluate and Display Inline

•	Context Panel: Conversions≻Column Vector

$M$ = $[- \frac{11}{5}, \frac{19}{5}, \frac{22}{5}]$ $\overset{to Vector}{\to}$ $[\begin{array}{c} - \frac{11}{5} \\ \frac{19}{5} \\ \frac{22}{5} \end{array}]$ $\overset{assign to a name}{\to}$ $m$

Obtain the equations of the lines that pass through the heads of $m \pm 4 u$ and that have direction V

$(m + 4 u) + s V$

$[\begin{array}{c} - \frac{11}{5} - \frac{64 \sqrt{266}}{399} - 7 s \\ \frac{19}{5} + \frac{46 \sqrt{266}}{399} - 11 s \\ \frac{22}{5} + \frac{58 \sqrt{266}}{399} + s \end{array}]$

$(m - 4 u) + s V$

$[\begin{array}{c} - \frac{11}{5} + \frac{64 \sqrt{266}}{399} - 7 s \\ \frac{19}{5} - \frac{46 \sqrt{266}}{399} - 11 s \\ \frac{22}{5} - \frac{58 \sqrt{266}}{399} + s \end{array}]$

Table 1.7.9(c) Implementation of the calculations listed in Table 1.7.9(b)

Maple Solution - Coded

•	Install the Student MultivariateCalculus package.

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

•	Define plane $P$ via the Plane command.

$P ≔ Plane (2 x - y + 3 z = 5) &colon;$

•	Define line $L$ via the Line command.

$L ≔ Line ([x = 2 - 3 t, y = 1 + 2 t, z = 3 + t]) &colon;$

•	Obtain N, the normal on plane $P$ via the GetNormal command.

$N ≔ GetNormal (P) &colon;$

•	Obtain point M, the intersection of line $L$ with plane $P$ , via the GetIntersection command.

$M ≔ GetIntersection (L, P) &colon;$

•	Obtain W, the direction of line $L$ , via the GetDirection command.

$W ≔ GetDirection (L) &colon;$

•	Obtain $Q$ , the plane containing point M, and the directions W and N, via the Plane command.

$Q ≔ Plane (M, W, N) &colon;$

•	Obtain V, the normal on plane $Q$ , via the GetNormal command.

$V ≔ GetNormal (Q) &colon;$

•	Get u, a unit vector along the line of intersection of planes $Q$ and $P$ by using the GetIntersection, GetDirection and Normalize commands.

$u ≔ Normalize (GetDirection (GetIntersection (Q, P))) &colon;$

•	Obtain m, the position vector to point M.

$m ≔ Vector (M) &colon;$

Obtain the equations of the lines that pass through the heads of $m \pm 4 u$ and that have direction V

$(m + 4 u) + s V$

$[\begin{array}{c} - \frac{11}{5} + \frac{64 \sqrt{266}}{399} + 7 s \\ \frac{19}{5} - \frac{46 \sqrt{266}}{399} + 11 s \\ \frac{22}{5} - \frac{58 \sqrt{266}}{399} - s \end{array}]$

$(m - 4 u) + s V$

$[\begin{array}{c} - \frac{11}{5} - \frac{64 \sqrt{266}}{399} + 7 s \\ \frac{19}{5} + \frac{46 \sqrt{266}}{399} + 11 s \\ \frac{22}{5} + \frac{58 \sqrt{266}}{399} - s \end{array}]$

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

$T ≔ solve (eval (2 x - y + 3 z = 5, [x = 2 - 3 t, y = 1 + 2 t, z = 3 + t])) &colon; M ≔ eval ([x = 2 - 3 t, y = 1 + 2 t, z = 3 + t], t = T) &colon; m ≔ eval (⟨x, y, z⟩, M) &colon;$

By inspection, write the vectors N and W

$N, W ≔ ⟨2, - 1, 3⟩, ⟨- 3, 2, 1⟩ &colon;$

Obtain $V = W \times N$ and $u = \frac{V \times N}{∥V \times N∥}$

$V ≔ CrossProduct (W, N) &colon;$

$u ≔ Normalize (CrossProduct (V, N)) &colon;$

Obtain the two lines $(m \pm 4 u) + s V$

$(m + 4 u) + s V$

$[\begin{array}{c} - \frac{11}{5} + \frac{64 \sqrt{266}}{399} + 7 s \\ \frac{19}{5} - \frac{46 \sqrt{266}}{399} + 11 s \\ \frac{22}{5} - \frac{58 \sqrt{266}}{399} - s \end{array}]$

$(m - 4 u) + s V$

$[\begin{array}{c} - \frac{11}{5} - \frac{64 \sqrt{266}}{399} + 7 s \\ \frac{19}{5} + \frac{46 \sqrt{266}}{399} + 11 s \\ \frac{22}{5} + \frac{58 \sqrt{266}}{399} - s \end{array}]$

Table 1.7.9(d) Vector-based calculation of the requisite lines.

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