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Chapter 1: Vectors, Lines and Planes
Section 1.7: Planes
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Example 1.7.3
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Obtain an equation for the plane that contains the points P:, and Q:, and that is parallel to the vector .
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Solution
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Mathematical Solution
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Figure 1.7.3(a) suggests how the requisite plane can be found. Position vectors P and Q, emanating from the origin O, respectively have their heads at points P and Q, which lie in the plane.
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Since the plane is parallel to the direction V, the vector V (in red) must lie in the plane. A second vector, , also lies in the plane, so the normal to the plane is .
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The equation of the plane now follows from the vector equation .
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Begin by obtaining the vectors
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Figure 1.7.3(a) Position vectors P and Q (blue), direction vector V (red), (black), and (green)
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= and = =
Conclude by implementing the vector equation of the plane, namely,
=
from which follows . A simpler form for the equation is clearly
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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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Write the sequence of points P and Q, and vector V.
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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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The traditional vector approach to finding this plane would be as follows.
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Define the position vectors P, Q, and R, and the direction vector V
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Context Panel: Assign to a Name≻P
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Context Panel: Assign to a Name≻Q
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Context Panel: Assign to a Name≻R
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Context Panel: Assign to a Name≻V
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Obtain the vector
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Context Panel: Assign Name
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Obtain the normal
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Context Panel: Assign Name
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Implement the vector equation of the line
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Common Symbols palette: Dot product operator
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Simplify the equation to the form by the obvious manipulations.
In agreement with what is stated in the Mathematical Solution, the two calculated vectors are
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=
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=
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The requisite plane can also be obtained by an algebraic calculation.
The general equation of the plane, namely, , must be satisfied by the coordinates of points P and Q; also, the normal (whose components are ) must be orthogonal to the vector V. These three conditions generate three equations in the four unknowns, namely, . Hence a solution for in terms of is possible, and leads to the desired equation of the plane with a judicious choice for . An interactive calculation along these lines is given below.
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Context Panel: Assign Name
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Write
Context Panel: Assign Function
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Write and press the Enter key.
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Context Panel: Solve≻Solve for Variables≻
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Context Panel: Assign to a Name≻
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Expression palette: Evaluation template
Press the Enter key
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Context Panel: Simplify≻Assuming Real
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Context Panel: Evaluate at a Point≻
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Maple Solution - Coded
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Initialize
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Install the Student MultivariateCalculus package.
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Define the position vectors P, Q, R, and the direction vector V
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Define the vectors P, Q, R, and V.
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Implement the vector equation for the plane
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<< Previous Example Section 1.7
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