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Chapter 1: Vectors, Lines and Planes
Section 1.4: Cross Product
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Example 1.4.5
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Find a unit vector orthogonal to the plane containing the points P:, Q:, R:.
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Solution
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Mathematical Solution
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The three blue vectors in Figure 1.4.5(a) are the position vectors to points P, Q, and R.
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The red and green vectors in the plane determined by these three points respectively represent the vectors
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and
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The gold vector N is a normal to the plane, and is obtained as the cross product
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use Student:-VectorCalculus, plots in
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local A,B,C,AB,AC,n,N,p1,p2,p3,R;
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A:=RootedVector(root=[0,0,0],<2,3,-1>):
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B:=RootedVector(root=[0,0,0],<5,-7,2>):
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C:=RootedVector(root=[0,0,0],<3,6,1>):
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R:=PositionVector([x,y,z]):
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AB:=RootedVector(root=A,convert(B-A,Vector)):
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AC:=RootedVector(root=A,convert(C-A,Vector)):
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n:=CrossProduct(AC,AB/6):
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N:=RootedVector(root=[2,3,-1],convert(n,Vector)):
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p1:=VectorCalculus:-PlotVector([AB,AC,N,A,B,C],color=[red,green,gold,blue,blue,blue],width=.25):
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p2:=implicitplot3d((R-A).convert(N,Vector),x=-1..8,y=-7..7,z=-5..5,style=surface,transparency=.8,color=red):
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p3:=display(p1,p2,scaling=constrained,axes=none,orientation=[180,-30,150],lightmodel=none,glossiness=0);
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Figure 1.4.5(a) Normal to plane determined by three given points
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= =
A unit vector collinear with N is then either one of
= =
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Maple Solution - Interactive
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Initialize
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Tools≻Load Package: Student Multivariate Calculus
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Loading Student:-MultivariateCalculus
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Define the position vectors P, Q, R
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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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Obtain the vector
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Common Symbols palette: Cross-product operator
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Context Panel: Assign Name
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Normalize
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Write N
Context Panel: Evaluate and Display Inline
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Context Panel: Student Multivariate Calculus≻Normalize
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=
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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
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Obtain and normalize the cross product of and
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<< Previous Example Section 1.4
Next Example >>
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