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 .