Chapter 1: Vectors, Lines and Planes
Section 1.7: Planes
Example 1.7.4
The planes S1: 3 x−7 y−9 z=8 and S2:5 x+4 y−2 z=6 intersect in a line L.
Find the parametric equations for L.
Find the equation of the plane that is perpendicular to L and that contains the point P:2,−3,1.
Solution
Mathematical Solution
Part (a)
The direction of line L is V=3−7−9×54−2 = |ijk3−7−954−2| = 50−3947, the cross product of the normals to planes S1 and S2.
A point on line L can be found by setting z=0 in the equations for planes S1 and S2 and solving for x and y. This gives the point 74/47,−22/47,0 so the vector form of the line is
xyz=74/47−22/470+t 50−3947
and the corresponding parametric equations are then
x=74/47+50 t,y=−22/47−39 t,z=47 t
Part (b)
The direction of line L, namely V, is the normal to the plane orthogonal to L. Hence, the vector form of the equation of this plane is R−P·V=0, or
(xyz−2−31)·50−3947 = 50x−2−39y+3+47z−1=0
which simplifies to 50 x−39 y+47 z=100+117+47=264.
Maple Solution - Interactive
Tools≻Load Package Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Define the planes S1 and S2
Control-drag the equation of each plane.
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 the line of intersection
Write the sequence of names S1 and S2
Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Intersection
Context Panel: Assign to a Name≻L
S1,S2 = << Plane 1 >>,<< Plane 2 >>→intersection<< Line 1 >>→assign to a nameL
Obtain the parametric equations for line L
Type L, the name of the line
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation≻parametric
L = << Line 1 >>→representationx=496623+50t,y=4333115−39t,z=−22793115+47t
An alternative algebraic approach also yields the parametric equations of L, albeit in a slightly different form. Simply solve the two equations defining planes S1 and S2 for any two of the unknowns x,y,z. The third unknown will then be the parameter along the line. As in the solution below, solving for x and y causes the parametrization to be x=xz,y=yz,z=z.
Form a sequence of the two equations defining planes S1 and S2
Context Panel: Solve≻Solve for Variables≻x,y
3 x−7 y−9 z=8,5 x+4 y−2 z=6→solve (specified)x=7447+5047z,y=−2247−3947z
The traditional vector-based solution obtains the direction for line L as the cross product of the normals to the two given planes. One point on the line of intersection is found by setting, say, z=0 in the equations for S1 and S2, and solving for the corresponding coordinates x and y.
Obtain the direction vector for the line as the cross product of normals
Common Symbols package: Cross product operator
3,−7,−9×5,4,−2 =
Obtain the coordinates of a point on the line of intersection
Form a list of the equations for S1 and S2. Press the Enter key.
Context Panel: Evaluate at a Point≻z=0
Context Panel: Solve≻Solve
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
The vector form for line L now follows as
xyz=14774−220+s 50−3947
as does the parametric form x=74/47+50 s,y=−22/47−39 s,z=47 s. This parametric form differs again from the two previous forms because the parameter along the line is yet again different.
Obtain the direction vector for L, which is the normal for the requisite plane
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Direction
Context Panel: Assign to a Name≻N
L = << Line 1 >>→direction →assign to a nameN
Obtain the equation of the plane
Form a sequence of the point P and the name of the normal N.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation
2,−3,1,N→make plane<< Plane 3 >>→representation50x−39y+47z=264
The vector-based approach to obtaining the plane orthogonal to L is as follows.
Define the position vectors P and R
Enter P as per Table 1.1.1.
Context Panel: Assign to a Name≻P
2,−3,1→assign to a nameP
Enter R as per Table 1.1.1.
Context Panel: Assign to a Name≻R
x,y,z→assign to a nameR
Implement the vector form for the equation of a plane
Common Symbols palette: Dot-product operator Press the Enter key.
R−P·N=0
50x−264−39y+47z=0
Maple Solution - Coded
L≔GetIntersectionPlane3 x−7 y−9 z=8,Plane5 x+4 y−2 z=6:
GetRepresentationL,form=parametric
x=496623+50t,y=4333115−39t,z=−22793115+47t
GetRepresentationPlane2,−3,1,GetDirectionL
50x−39y+47z=264
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