Chapter 1: Vectors, Lines and Planes
Section 1.4: Cross Product
Example 1.4.5
Find a unit vector orthogonal to the plane containing the points P:1,2,−1, Q:3,1,2, R:4,−5,1.
Solution
Mathematical Solution
The three blue vectors in Figure 1.4.5(a) are the position vectors to points P, Q, and R.
The red and green vectors in the plane determined by these three points respectively represent the vectors
Q−P=2−13 and R−P=3−72
The gold vector N is a normal to the plane, and is obtained as the cross product
use Student:-VectorCalculus, plots in
module()
local A,B,C,AB,AC,n,N,p1,p2,p3,R;
A:=RootedVector(root=[0,0,0],<2,3,-1>):
B:=RootedVector(root=[0,0,0],<5,-7,2>):
C:=RootedVector(root=[0,0,0],<3,6,1>):
R:=PositionVector([x,y,z]):
AB:=RootedVector(root=A,convert(B-A,Vector)):
AC:=RootedVector(root=A,convert(C-A,Vector)):
n:=CrossProduct(AC,AB/6):
N:=RootedVector(root=[2,3,-1],convert(n,Vector)):
p1:=VectorCalculus:-PlotVector([AB,AC,N,A,B,C],color=[red,green,gold,blue,blue,blue],width=.25):
p2:=implicitplot3d((R-A).convert(N,Vector),x=-1..8,y=-7..7,z=-5..5,style=surface,transparency=.8,color=red):
p3:=display(p1,p2,scaling=constrained,axes=none,orientation=[180,-30,150],lightmodel=none,glossiness=0);
print(p3);
end module:
end use:
Figure 1.4.5(a) Normal to plane determined by three given points
N=Q−P×R−P= |ijk2−133−72| = 195−11
A unit vector collinear with N is then either one of
±NN = ±1192+52+−112195−11 = ±1133195−11
Maple Solution - Interactive
Initialize
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Define the position vectors P, Q, R
Enter P as per Table 1.1.1.
Context Panel: Assign to a Name≻P
1,2,−1→assign to a nameP
Enter Q as per Table 1.1.1.
Context Panel: Assign to a Name≻Q
3,1,2→assign to a nameQ
Enter R as per Table 1.1.1.
Context Panel: Assign to a Name≻R
4,−5,1→assign to a nameR
Obtain the vector N=Q−P×R−P
Common Symbols palette: Cross-product operator
Context Panel: Assign Name
N=Q−P×R−P→assign
Normalize N
Write N Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Normalize
N = →normalize
Maple Solution - Coded
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
P,Q,R≔1,2,−1,3,1,2,4,−5,1:
Obtain and normalize the cross product of Q−P and R−P
Apply the CrossProduct and Normalize commands.
NormalizeCrossProductQ−P,R−P
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