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Chapter 1: Vectors, Lines and Planes
Section 1.7: Planes
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Example 1.7.8
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Obtain an equation for , the line that passes through the point A:, intersects L, the line , and is parallel to P, the plane .
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Solution
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Mathematical Solution
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The search for line begins with a "diagnosis" in which a strategy for its determination is proposed.
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Strategy
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As per Figure 1.7.8(a), let (pink) be a plane through point A (green) and parallel to plane (gray).
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Let B (black) be the point of intersection of line L (red) with plane Q.
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The line through A and B is the line .
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Figure 1.7.8(a) Planes and , points A and B, and lines and
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Implementation
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As a first step, plane must be found. Since it is parallel to the given plane , it has the same normal, N. Hence, plane is given by , that is, by
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The intersection of line with plane is found by solving , that is,
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for so point B is then = .
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Finally, , the line through both A and B, is given by , that is, by
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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 plane
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Control-drag the equation of 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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Define the line
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Using the technique of control-drag, make 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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Obtain N, the normal for plane
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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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Obtain plane
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Form a sequence of point A and the name of 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 point B, the intersection of line and plane
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Write the sequence of names for line and plane .
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Context Panel: Student Multivariate Calculus≻Lines & Planes≻Intersection
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Context Panel: Assign to a Name≻B
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Obtain line
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Write a sequence of point A and the name of point B.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Line
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation≻vectors
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Maple Solution - Coded
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Table 1.7.8(a) uses the command-form of the "Lines & Planes" tools in the Student MultivariateCalculus package to obtain an equation for line .
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Use the Plane command to define plane .
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Use the Line command to define line .
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Use the GetNormal command to obtain N, the normal on plane .
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Use the Plane command to obtain , the plane parallel to and through A.
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Use the GetIntersection command to obtain B, the intersection of line with plane .
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Use the Line command to obtain , the line through points A and B.
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Table 1.7.8(a) Command-based determination of line
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