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 Tips and Techniques: Formulas for Dot and Cross ProductsRobert J. LopezEmeritus Professor of Mathematics and Maple FellowMaplesoft
<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field>Early in the multivariate calculus course, or even before that in a general physics course, formulas for computing both the dot product and the cross-product appear. Typically, these formulas are presented as 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for the dot product 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for the cross-product 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. Each such formula shows how to compute the appropriate result, but gives no insight into the meaning or utility of the computed quantity. To remedy this, the interpretative 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are generally given.In the case of the cross-product, the typical presentation states that the cross-product 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 is orthogonal to both A and B, and points in a direction consistent with the right-hand rule.In this article, we will show how Maple can be used to provide some additional insight into both concepts.
<Text-field style="Heading 1" layout="Heading 1">Initializations</Text-field>restart;with(Student[LinearAlgebra]):
infolevel[Student[LinearAlgebra]] := 1:
with(plots):v1 := <2,2,2>:
v2 := <3,2,4>:
v3 := v2.v1/Norm(v1,2)^2*v1:
v4 := v2-v3:
p1 := display(arrow([0,0,0],<2,2,2>,color=green),transparency=.5):
p2 := arrow([0,0,0],<3,2,4>,color=red,width=.1, head_length=.4):
p3 := arrow([0,0,0],1.2*v3,color=black,width=.1, head_length=.4):
p4 := arrow(v2,-v4, color=blue,width=.1):
Fig1 := display([p||(1..4)],scaling=constrained, axes=box, labels=[x,y,z], orientation=[15,75]):
<Text-field style="Heading 1" layout="Heading 1">The Dot Product</Text-field>For the dot product 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, one can trivially show that 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by using the vectorsA := <seq(a[k],k=1..3)>:
B := <seq(b[k],k=1..3)>:
A,B;The left side of the relation is obtained in Maple withA.B;while the right side is obtained withNorm(A,2)*Norm(B,2)*cos(VectorAngle(A,B));Unfortunately, this demonstration is trivial because the cosine of the angle between the vectors A and B is computed by the very formula we purport to demonstrate. What we need is an independent way of determining the angleLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYsNiVRKCZ0aGV0YTtGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRic=.For this, consider Figure 1, a representation of two vectors A and B, along with the projection of A along the direction of B, and the negative of the component of A orthogonal to B.
Fig1;
Figure 1 Vector A (red) projected onto vector B (green).Direction of B indicated by black arrow.In blue, 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The right triangle so formed determines 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
In the right triangle formed by A (in red), its projection along the black extension of B (in green) and the orthogonal component (in blue), we 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where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2J1EhRicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGNy1JJW1zdWJHRiQ2JS1GLDYnUSJBRidGL0YyRjVGOC1GLDYnUSJCRidGL0YyRjVGOC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUYsNiVGLi9GM0YxL0Y2USdpdGFsaWNGJw== is the vector projection of A on B.With an appropriate convention on the domain for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYsNiVRKCZ0aGV0YTtGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRic=, the absolute value on 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 in the rightmost numerator can be removed, and we have 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Unfortunately, projection of vectors typically uses the dot product in its computations, so the lessons on the dot product usually preceed the lessons on projection. Here's a way to establish the relationship for 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 without having to resort to results based on the use of the dot product.Motivated by the right triangle shown in Figure 1, let 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 be the line that passes through the origin and has direction BLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbW9HRiQ2MFEiLkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRJDBlbUYnLyUncnNwYWNlR0ZGLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYn Then, the component of A along B can be found by minimizing the functionf := collect(Norm(t*B-A,2)^2,t);where we choose to minimize the square of the distance from the tip of A to the line containing B. This is a one-dimensional minimization that yields to T := solve(diff(f,t) = 0,t);The astute reader will recognize this as 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 although this representation is not necessary for writing LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtRiw2JVEkY29zRicvRjBRJmZhbHNlRicvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2MFEwJkFwcGx5RnVuY3Rpb247RidGPC8lJmZlbmNlR0Y7LyUqc2VwYXJhdG9yR0Y7LyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJWZvcm1HUSZpbmZpeEYnLyUnbHNwYWNlR1EkMGVtRicvJSdyc3BhY2VHRlUvJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5RictSShtZmVuY2VkR0YkNiMtRiM2Iy1GLDYlUSgmdGhldGE7RidGOkY8Ris= assimplify(Norm(T*B,2)/Norm(A,2)) assuming real;which is just the length of the adjacent side over the length of the hypotenuse in the right triangle shown in Figure 1.Thus, we have again 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 without the need for a projection formula based on the dot product.
<Text-field style="Heading 1" layout="Heading 1">The Cross-Product</Text-field>The cross-product of two vectors 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 and 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 is generally given by the 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supported by the 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Normally, these statements are then followed by a declaration the properties of the cross-product vector C, namely,
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, B, and C form a right-handed system of vectors (i.e., obey the right-hand rule)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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYsNiVRKCZ0aGV0YTtGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRic= is the angle between A and BTable 1 Properties of the cross-product
Of course, these properties are easily verified, especially in Maple. For example, if we defineC := A &x B;then it is immediate to showA.C = simplify(A.C);
B.C = simplify(B.C);Probably the easiest way to verify the right-hand rule is via an example such as the one captured in Figure 2.
CrossProductPlot(<1,2,-1>,<-3,2,-1>, axes=box, vectorcolors=[red, blue, black]);Figure 2 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 (in black) for A = 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 (in red) and B = 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 (in blue)
The third property in Table 1 can be verified if we can derive an expression for the sine of the angle between A and B. Again using the right triangle in Figure 1, we obtain the blue vector as the negative of the component of A orthogonal to B. This is given by the vector subtractionV := -simplify(A - T*B);Hence, 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 is given bySIN := simplify(Norm(V,2)/Norm(A,2)) assuming real;That this is 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 can be verified by showing the numerator is actually 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 by means of the calculationNorm(A &x B,2);Hence, we have 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If we define a vector C asC := <seq(c[k],k=1..3)>;we can actually derive a formula for the cross-product from the three properties in Table 1. The orthogonality expressed by the first property in Table 1 is captured in the equationsq1 := A.C = 0;
q2 := B.C = 0;whereas the third property is captured by the equationq3 := Norm(C,2) = Norm(A,2)*Norm(B,2)*SIN;Solving for the components of the vector C, we haveS := solve({q1,q2,q3},{seq(c[k],k=1..3)});from which we determine the two possible solutions for C, namely,V1 := eval(C,S[1]):
V2 := eval(C,S[2]):
V1,V2;That these two solutions are the negatives of each other is established byV1 + V2;We leave it to the reader to select the unique solution that satisfies the right-hand rule.Legal Notice: The copyright for this application is owned by Maplesoft. The application is intended to demonstrate the use of Maple to solve a particular problem. It has been made available for product evaluation purposes only and may not be used in any other context without the express permission of Maplesoft. 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