Chapter 1: Vectors, Lines and Planes
Section 1.7: Planes
Example 1.7.6
Find the distance from the point P:2,−3,1 to the plane 5 x−7 y+9 z=11.
Solution
Mathematical Solution
Figure 1.7.6(a) shows plane S, the arbitrary point A on S, and the point P.
The red arrow represents the normal N, while the green arrow represents the vector W=P−A, the vector from A to P.
The black arrow represents WN, the component of W along N. The magnitude of WN is the distance from point P to the plane S.
The arbitrary point A has coordinates 0,0,11/9, obtained by setting x=y=0 in the equation for S and solving the resulting equation for z=11/9.
Figure 1.7.6(a) Plane S, its normal N (red), points P and A, the vectors W=P−A (green) and WN (black)
The essential calculations are
W=2−31−0011/9 = 2−3−2/9 and WN=W·NN·N = 10+21−225+49+81 = 29155
where WN is the scalar projection (Table 1.3.1) of W on N.
Maple Solution - Interactive
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Define the given plane and assign it the name S
Control-drag the equation of the plane.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Plane
Context Panel: Assign to a Name≻S
5 x−7 y+9 z=11→make plane<< Plane 1 >>→assign to a nameS
Obtain the distance from point P to the plane S
Write a sequence of the point P (as a list) and S, the name of the plane.
Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Distance
Context Panel: Approximate≻5 (digits)
2,−3,1,S = 2,−3,1,<< Plane 1 >>→distance29155155→at 5 digits2.3294
An implementation of the traditional vector-based calculation is given below in Table 1.7.6(a).
Enter P as per Table 1.1.1.
Context Panel: Assign to a Name≻P
2,−3,1→assign to a nameP
Enter A as per Table 1.1.1.
Context Panel: Assign to a Name≻A
0,0,11/9→assign to a nameA
Context Panel: Assign Name
W=P−A→assign
Obtain N, the normal to plane S
Type S, the name of the plane. Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Normal
Context Panel: Assign to a Name≻N
S = << Plane 1 >>→normal →assign to a nameN
Obtain WN, the magnitude of the projection of W on N
Common Symbols palette: Dot product operator
W·NN·N = 29155155→at 5 digits2.3294
Alternate calculation of the magnitude of the scalar projection of W on N
Write the sequence of vectors W and N. Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Projection
Context Panel: Norm≻Euclidean
W,N = →projection →Euclidean-norm29155155
Table 1.7.6(a) Traditional vector-based calculation of distance from point to plane
Maple Solution - Coded
Table 1.7.6(b) calculates the distance from point P to plane S via a command-based implementation of the "Lines & Planes" tools in the Student MultivariateCalculus package.
Apply the Distance and Plane commands.
Obtain a five-digit floating-point (decimal) value with the evalf command.
d≔Distance2,−3,1,Plane5 x−7 y+9 z=11 = 29155155
evalfd,5 = 2.3294
Table 1.7.6(b) Command-based implementation of "Lines & Planes" tools
Table 1.7.6(c) obtains the distance from point P to plane S via a command-based implementation of the traditional vector calculation.
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define the vectors P, N, and A.
P,N,A≔2,−3,1,5,−7,9,0,0,11/9:
Apply the DotProduct and Norm commands to obtain the scalar projection of P−A on N.
DotProductP−A,NNormN = 29155155
Table 1.7.6(c) Command-based implementation of the traditional vector calculation
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