The essential step of adding and subtracting the "right" expression in the calculations in the Mathematical Solution was first discovered by working backwards in Maple. The "magic" by which this discovery was made is articulated below.
Initialize
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Tools≻Load Package: Student Multivariate Calculus
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Loading Student:-MultivariateCalculus
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Context Panel: Assign to a Name≻A
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Context Panel: Assign to a Name≻B
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Compute and expand the square of its norm
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Common Symbols palette: Cross-product operator
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Typeset the square of the norm.
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Apply the expand command and assign the result to the name
Press the Enter key.
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Obtain the intended right-hand side:
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Typeset the square of the norms of A and B.
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Apply the Angle command to obtain the angle
Press the Enter key.
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Context Panel: Simplify≻Simplify
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If the factors in the first form of the right-hand side are multiplied out, the result is
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which is nothing more than − . This is what the expression must become if is to be recognized as the square of . However, does not immediately factor into this form, and the reason why not has to be determined. To this end, split into its positive and negative terms. This is done with an application of the selectremove command that returns terms that "have" a given subexpression, and also returns the terms that don't. (The has command returns true or false, either the subexpression is there or it isn't.)
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The expression has been split into , its negative terms, and , its positive terms. The positive terms should factor to but they don't. The negative terms should factor to , but they don't either. Why not?
Consider the difference between and :
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The three terms in must be added to for the positive terms in to factor to . Indeed, this can be further verified by factoring the sum of and :
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The three terms in must be subtracted from for the negative terms in to factor to . Indeed, this can be further verified by factoring the difference of and :
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(Note that the three terms in are precisely the three red terms added and subtracted in the Mathematical Solution!)
Consequently, the calculations in the Mathematical Solution can be implemented, and the desired result obtained. The algebraic manipulations needed to discover how to proceed are the forte of Maple.