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Chapter 1: Vectors, Lines and Planes
Section 1.7: Planes
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Example 1.7.6
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Find the distance from the point P: to the plane .
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Solution
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Mathematical Solution
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Figure 1.7.6(a) shows plane , the arbitrary point A on , and the point P.
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The red arrow represents the normal N, while the green arrow represents the vector , the vector from A to P.
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The black arrow represents , the component of W along N. The magnitude of is the distance from point P to the plane .
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The arbitrary point A has coordinates , obtained by setting in the equation for and solving the resulting equation for .
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Figure 1.7.6(a) Plane , its normal N (red), points P and A, the vectors (green) and (black)
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The essential calculations are
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= and = =
where is the scalar projection (Table 1.3.1) of W on N.
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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 the given plane and assign it the name
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Control-drag the equation of the plane.
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Context Panel: Student Multivariate Calculus≻Lines & Planes≻Plane
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Context Panel: Assign to a Name≻
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Obtain the distance from point P to the plane
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Write a sequence of the point P (as a list) and , the name of the plane.
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Context Panel: Evaluate and Display Inline
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Context Panel: Student Multivariate Calculus≻Lines & Planes≻Distance
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Context Panel: Approximate≻5 (digits)
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=
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An implementation of the traditional vector-based calculation is given below in Table 1.7.6(a).
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Context Panel: Assign to a Name≻P
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Context Panel: Assign to a Name≻A
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Context Panel: Assign Name
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Obtain N, the normal to plane
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Type , the name of the plane.
Context Panel: Evaluate and Display Inline
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Context Panel: Student Multivariate Calculus≻Lines & Planes≻Normal
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Context Panel: Assign to a Name≻N
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=
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Obtain , the magnitude of the projection of W on N
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Common Symbols palette: Dot product operator
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Context Panel: Evaluate and Display Inline
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Context Panel: Approximate≻5 (digits)
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=
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Alternate calculation of the magnitude of the scalar projection of W on N
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Write the sequence of vectors W and N.
Context Panel: Evaluate and Display Inline
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Context Panel: Student Multivariate Calculus≻Lines & Planes≻Projection
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Context Panel: Norm≻Euclidean
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=
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Table 1.7.6(a) Traditional vector-based calculation of distance from point to plane
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Maple Solution - Coded
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Table 1.7.6(b) calculates the distance from point P to plane via a command-based implementation of the "Lines & Planes" tools in the Student MultivariateCalculus package.
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Obtain a five-digit floating-point (decimal) value with the evalf command.
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=
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Table 1.7.6(b) Command-based implementation of "Lines & Planes" tools
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Table 1.7.6(c) obtains the distance from point P to plane via a command-based implementation of the traditional vector calculation.
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Install the Student MultivariateCalculus package.
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Define the vectors P, N, and A.
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Apply the DotProduct and Norm commands to obtain the scalar projection of on N.
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=
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Table 1.7.6(c) Command-based implementation of the traditional vector calculation
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