Chapter 1: Vectors, Lines and Planes
Section 1.7: Planes
Example 1.7.13
Obtain an equation for P, the plane that contains L, the line
x=3+t,y=2+3 t,z=4− t
and whose distance from the point A:2,−1,3 is a maximum.
Solution
Mathematical Solution
The schematic in Figure 1.7.13(a) depicts line L, point A, the line connecting point A to the projection of A on L, and plane P.
The view of line L is "down the barrel" so that the line is orthogonal to the plane of the figure. The gold dot is then all that is visible of line L.
Plane P is therefore seen as the green line. Since P must contain L, it is free to rotate about L in such a way as to maximize its distance from A.
Point A is represented by the black square. Its projection onto P is point B, represented by the black dot. The length of the red line segment connecting A and B is then the distance from A to P.
Figure 1.7.13(a) Schematic: Projection of A onto L
The distance from A to P is a maximum when B is the foot of the projection of A onto L. A vector from A to B is therefore N, the normal to P, and P is determined by N and point B.
The orientation of plane P as it rotates around line L is controlled by the slider beneath the figure.
Point B is the point on L that is closest to A, so it can be found by minimizing the distance from A to L.
The square of d, the distance from A to L, is given by
d2=3+t−22+2+3 t+12+4−t−32=11+6 t+11 t2
This distance is minimized by setting the derivative equal to zero and solving for t^=−9/11, giving 24/11,−5/11,53/11 for the coordinates of point B.
The normal for plane P is then
N=B−A = 11124−553−2−23 = 2111310
so the equation for P is then
R−B·N = xyz−11124−553·2111310 = 211x+3 y+10 z−49=0
from which P can be represented by the equation x+3 y+10 z=49.
Maple Solution - Interactive
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Define line L
Form 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=3+t,y=2+3 t,z=4− t→make line<< Line 1 >>→assign to a nameL
Define point A
Write the list representing point A.
Context Panel: Assign to a Name≻A
2,−1,3→assign to a nameA
Obtain point B, the projection of point A onto line L
Write the sequence of names for point A and line L.
Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Projection
Context Panel: Assign to a Name≻B
A,L = 2,−1,3,<< Line 1 >>→projection2411,−511,5311→assign to a nameB
Obtain N, the normal for plane P, as the vector from point A to point B
Write the difference B−A.
Context Panel: Conversions≻Column Vector
Context Panel: Assign to a Name≻N
B−A = 211,611,2011→to Vector2116112011→assign to a nameN
Obtain plane P as the plane that has normal N and that contains point B
Write the sequence of names for point B and normal vector N.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Plane
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation
B,N→make plane<< Plane 1 >>→representation2x11+6y11+20z11=9811
Of course, a simpler form for the equation of plane P is x+3 y+10 z=49.
Maple Solution - Coded
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Apply the Line command to define line L.
L≔Linex=3+t,y=2+3 t,z=4− t:
Define point A.
A≔2,−1,3:
Apply the Projection command to obtain B, the projection of A on L.
B≔ProjectionA,L:
Use the Vector command to define N, the vector from A to B.
N≔VectorB−A:
Use the Plane command to define P, the plane whose normal is N and which contains B.
P≔PlaneB,N:
Apply the GetRepresentation command to plane P to obtain its equation.
GetRepresentationP
2x11+6y11+20z11=9811
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