Dot and Cross Products Related to Complex Multiplication

A complex number can be considered as a vector and vice versa, both points of view having their own context. The operations transforming vectors and complex numbers are particular to them; vectors use the dot and cross products while complex numbers use multiplication and conjugation (written using an overbar). How are these two pairs of operations related to one another when their operands are identified either as vectors or as complex numbers?
This Demonstration shows the equation relating the two kinds of operations (writing complex numbers in uppercase letters, and , and their corresponding vectors as and ).
Some possible questions to address are:
When are and perpendicular? (answer: when );
When are and parallel? (answer: when —the usual test cannot be used if );
More generally, what is the value of the area of the parallelogram with sides and ? (answer: ).
When is ? (answer: when is a real number).
Prove that when , then .
Prove that the triangle with side lengths , , and is a right triangle.
Show that the magnitude of the projection of onto (the length of the side of the red triangle that is parallel to ) is equal to .


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