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Chapter 1: Vectors, Lines and Planes
Section 1.7: Planes
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Example 1.7.11
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If plane P, and lines and , are given respectively by
find an equation for any line that is parallel to P, and intersects the lines in such a way as to form, between them, a segment of length .
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Solution
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Mathematical Solution
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A vector V from to and N, the normal to P, are
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Line is parallel to P if , and the condition on the length of the segment between and is , or equivalently,
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These two conditions give two equations in and , thus determining any line satisfying these conditions. Two such lines exist, one (red, in Figure 1.7.11(a)) actually lying in plane P, and the other (green) above the plane.
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>
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use plots in
module()
local P,L1,L2,V,N,Ans1,Ans2,p1,p2,p3,p4,p5,p6;
P:=x-y+z=6;
L1:=<2*t+1,t-2,2-2*t>;
L2:=<2-m,3+m,4+m>;
V:=L1-L2;
N:=<1,-1,1>;
Ans1:=<-1-2*s,-3-s,4+s>;
Ans2:=<5+s,-s,-2-2*s>;
p1:=implicitplot3d(P,x=-5..10,y=-5..4,z=-5..8,style=surface,color=pink,transparency=.5);
p2:=spacecurve(Ans1,s=-5..2,color=red,thickness=3,numpoints=2);
p3:=spacecurve(Ans2,s=-4..1,color=green,thickness=3,numpoints=2);
p4:=spacecurve(L1,t=-3..3,color=black,thickness=3,numpoints=2);
p5:=spacecurve(L2,m=-6..2,color=blue,thickness=3,numpoints=2);
p6:=display([p1,p2,p3,p4,p5],scaling=constrained,axes=none,orientation=[-35,-45,25]);
print(p6);
end module:
end use:
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Figure 1.7.11(a) Lines (black), (blue), plane P, and two solutions for , in red and green.
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The conditions and lead to the equations
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whose solutions are and . Hence there are two possible lines, namely,
and
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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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For , define lines as the position vectors
Define N, the normal for plane
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Context Panel: Assign to a Name≻L[1]
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Context Panel: Assign to a Name≻L[2]
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Context Panel: Assign to a Name≻N
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Obtain V, a vector from to
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Write
Context Panel: Assign Name
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Impose the conditions and and solve the resulting equations
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Write the sequence
Press the Enter key.
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Context Panel: Solve≻Solve
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Context Panel: Assign to a Name≻S
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In terms of the parameter , obtain line as a position vector
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Context Panel: Evaluate and Display Inline
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=
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Context Panel: Evaluate and Display Inline
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=
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Maple Solution - Coded
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Install the Student MultivariateCalculus package.
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For , define lines as the position vectors
Define N, the normal for plane and define
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Via the Norm, DotProduct and solve commands:
Impose and solve the equations arising from the conditions and
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Use the eval command to obtain line L
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Use the eval command to obtain line L
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