Chapter 1: Vectors, Lines and Planes
Section 1.7: Planes
Example 1.7.10
Obtain an equation for L, the line that is parallel to P and Q, the planes
2 x−7 y+9 z=12 and 3 x+5 y−8 z=1
and intersects L1 and L2, the lines
xyz=123+u2−11 and xyz=2−12+v1−32
Solution
Mathematical Solution
For line λ to be parallel to planes P and Q, it has to have the same direction as the line of intersection of P and Q.
This can be seen in Figure 1.7.10(a) where line λ, drawn in green, is parallel to the intersection of planes P and Q. (Rotate the figure to see this is so.)
Let V be the direction of λ. Then a vector from L1 to L2 is given by L2−L1. This vector must be parallel to V. Hence, solve L2−L1×V=0 for u,v=u^,v^.
λ can then be given by R=L1x=a|f(x)u=u^+p V.
use plots in module() local P,Q,S,V,L1,L2,q,L,p1,p2,p3,p4,p5; P:=2*x-7*y+9*z=12: Q:=3*x+5*y-8*z=1:
S:=solve({P,Q},{x,y}):
V:=map(coeff,eval(<x,y,z>,S),z):
L1,L2:=<2*u+1,2-1*u,3+u>,<2+v,-1-3*v,2+2*v>:
q:=solve(Equate(L1-L2,k*V),{u,v,k}):
L:=eval(L1,q)-s*eval(k*V,q):
p1:=implicitplot3d([P,Q],x=-2..5,y=-3..5,z=-1..5,style=surface,color=[red,gold]): p2:=spacecurve(L1,u=-1..2,color=black,thickness=3,numpoints=2): p3:=spacecurve(L2,v=-1..1,color=blue,thickness=3,numpoints=2): p4:=spacecurve(L,s=0..1,color=green,thickness=5,numpoints=2): p5:=display([p1,p2,p3,p4],scaling=constrained,axes=none,orientation=[-175,75,-75],lightmodel=none): print(p5); end module: end use:
Figure 1.7.10(a) Planes P and Q (red and gold, respectively); lines L1 and P2 (black and blue, respectively); line λ (green)
The vector V can be obtained as V=NP×NQ, where NP and NQ are normals for planes P and Q, respectively. It can also be obtained via the "Lines & Planes" tools in the Student MultivariateCalculus package.
Maple Solution - Interactive
Define lines L1 and L2 as position vectors L1 and L2
Context Panel: Assign to a Name≻L[1]
1,2,3+u <2,−1,1>→assign to a nameL1
Context Panel: Assign to a Name≻L[2]
2,−1,2+v <1,−3,2>→assign to a nameL2
Define NP and NQ, normals to planes P and Q, respectively
Context Panel: Assign to a Name≻N[P]
2,−7,9→assign to a nameNP
Context Panel: Assign to a Name≻N[Q]
3,5,−8→assign to a nameNQ
Obtain V, the direction vector for the line of intersection of planes P and Q
Write V=NP×NQ for the direction of line λ. Context Panel: Assign Name
V=NP×NQ→assign
Solve the equations L2−L1× V=0 for u,v
Write L2−L1× V and press the Enter key.
Context Panel: Conversions≻To List
Context Panel: Solve≻Solve
Context Panel: Assign to a Name≻S
L2−L1× V
→to list
−50−179v+74u,51u−42−9v,76+76v−97u
→solve
u=7691,v=691
→assign to a name
S
Obtain line λ
Expression palette: Evaluation template Evaluate L1+p V at u=u^.
x,y,z=L1+p Vx=a|f(x)S
Maple Solution - Coded
The following calculations obtain line λ via the "Lines & Planes" tools of the Student MultivariateCalculus package. Collinearity of the direction vectors for the line of intersection of planes P and Q and the line through generic points on lines L1 and L2 can be established either by showing their cross product is the zero vector, or by showing that the vectors are proportional. The first approach is taken in the Interactive Solution above; the second, in the Coded Solution below.
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Use the Plane command to define plane P.
P≔Plane2 x−7 y+9 z=12:
Use the Plane command to define plane Q.
Q≔Plane3 x+5 y−8 z=1:
Use the GetIntersection command to obtain the line of intersection of planes P and Q.
Use the GetDirection command to get the direction vector for the line of intersection.
V≔GetDirectionGetIntersectionP,Q:
Define A, the generic point on line L1.
A≔1+2 u,2−u,3+u:
Define B, the generic point on line L2.
B≔2+v,−1−3 v,2+2 v:
Use the Line command to obtain the line from A to B.
Use the GetDirection command to obtain W, the direction vector for the line through A and B.
W≔GetDirectionLineA,B:
Use the Equate command to equate corresponding components of W and a multiple of V.
Use the solve command to solve for u,v,k.
S≔solveEquateW,k V
k=−591,u=7691,v=691
Use the eval command to evaluate the generic point A at the solution S.
Write the vector form of the line with direction V that intersects lines L1 and P2.
x,y,z=VectorevalA,S+p V
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