Chapter 1: Vectors, Lines and Planes
Section 1.7: Planes
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Example 1.7.13
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Obtain an equation for P, the plane that contains L, the line
and whose distance from the point A: is a maximum.
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Solution
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Mathematical Solution
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The schematic in Figure 1.7.13(a) depicts line , point A, the line connecting point A to the projection of A on , and plane .
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The view of line 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 .
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Plane is therefore seen as the green line. Since must contain , it is free to rotate about in such a way as to maximize its distance from A.
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Point A is represented by the black square. Its projection onto 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 .
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Figure 1.7.13(a) Schematic: Projection of A onto
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The distance from A to is a maximum when B is the foot of the projection of A onto . A vector from A to B is therefore N, the normal to , and is determined by N and point B.
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The orientation of plane as it rotates around line is controlled by the slider beneath the figure.
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Point B is the point on that is closest to A, so it can be found by minimizing the distance from A to .
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The square of , the distance from A to , is given by
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This distance is minimized by setting the derivative equal to zero and solving for , giving for the coordinates of point B.
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The normal for plane is then
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= =
so the equation for is then
= =
from which can be represented by the equation .
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Maple Solution - Interactive
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Tools≻Load Package: Student Multivariate Calculus
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Loading Student:-MultivariateCalculus
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Define line
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Form a list of the parametric equations for line .
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Context Panel: Student Multivariate Calculus≻Lines & Planes≻Line
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Context Panel: Assign to a Name≻
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Define point A
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Write the list representing point A.
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Context Panel: Assign to a Name≻A
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Obtain point B, the projection of point A onto line
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Write the sequence of names for point A and line .
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Context Panel: Evaluate and Display Inline
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Context Panel: Student Multivariate Calculus≻Lines & Planes≻Projection
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Context Panel: Assign to a Name≻B
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Obtain N, the normal for plane , as the vector from point A to point B
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Write the difference .
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Context Panel: Evaluate and Display Inline
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Context Panel: Conversions≻Column Vector
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Context Panel: Assign to a Name≻N
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Obtain plane as the plane that has normal N and that contains point B
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Write the sequence of names for point B and normal vector N.
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Context Panel: Student Multivariate Calculus≻Lines & Planes≻Plane
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Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation
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Of course, a simpler form for the equation of plane is .
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Maple Solution - Coded
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Install the Student MultivariateCalculus package.
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Apply the Line command to define line .
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Apply the Projection command to obtain B, the projection of A on .
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Use the Vector command to define N, the vector from A to B.
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Use the Plane command to define , the plane whose normal is N and which contains B.
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Apply the GetRepresentation command to plane to obtain its equation.
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