Chapter 1: Vectors, Lines and Planes
Section 1.7: Planes
Example 1.7.7
Find the distance between the parallel planes 2 x+3 y+4 z=5 and 2 x+3 y+4 z=7.
Solution
Mathematical Solution
Figure 1.7.7(a) shows the parallel planes S1 and S2.
The red arrow represents the common normal N; the green arrow, the vector W=B−A, where A and B are position vectors to the arbitrary points A and B, respectively; and the black arrow represents WN, the component of W along N.
The arbitrary points on the two planes are taken as
A:0,0,5/4 and B:0,0,7/4
The vectors N and W are then
Figure 1.7.7(a) Distance between parallel planes: d=WN
N=234 and W=B−A=007/4−005/4=001/2
so
d=WN = W·NN·N = 24+9+16=229
Maple Solution - Interactive
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Define the two planes and name them S1 and S2, respectively
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
2 x+3 y+4 z=5→make plane<< Plane 1 >>→assign to a nameS1
2 x+3 y+4 z=7→make plane<< Plane 2 >>→assign to a nameS2
Obtain the distance between the two planes
Form the sequence of the two names of the planes. Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Distance
Context Panel: Approximate≻5 (digits)
S1,S2 = << Plane 1 >>,<< Plane 2 >>→distance22929→at 5 digits0.37140
The traditional vector-based calculation relies on projection, and is implemented in Table 1.7.7(a), below.
Define position vectors A and B, and the common normal N
Enter A as per Table 1.1.1.
Context Panel: Assign to a Name≻A
0,0,5/4→assign to a nameA
Enter B as per Table 1.1.1.
Context Panel: Assign to a Name≻B
0,0,7/4→assign to a nameB
Enter N as per Table 1.1.1.
Context Panel: Assign to a Name≻N
2,3,4→assign to a nameN
Obtain W=B−A, a vector from point A to point B , and obtain its scalar projection on N
Context Panel: Assign Name
W=B−A→assign
Common Symbols palette: Dot product operator
Context Panel: Evaluate and Display Inline
W·NN·N = 22929→at 5 digits0.37140
Table 1.7.7(a) Traditional vector-based calculation of distance between parallel planes
Maple Solution - Coded
Having defined the planes Sk,k=1,2, within the framework of the "Lines & Planes" tools of the Student MultivariateCalculus package, the distance between the planes is obtained with the Distance command.
DistanceS1,S2 = 22929
With these same tools, the traditional vector-based calculation is implemented in Table 1.7.7(b). The GetPoint command is used to return an arbitrary point on each plane; the convert command changes the resulting list to a Vector, which is then projected onto the normal, N, via the Projection command. The GetNormal command returns N, the common normal. The Norm command returns the length of the projection, which is the requisite distance.
WN≔ProjectionconvertGetPointS2−GetPointS1,Vector,GetNormalS1
NormWN = 22929
Table 1.7.7(b) Traditional vector-based calculation via the "Lines & Planes" tools
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