Illicit expressions in vector algebra

In vector geometry there are 2 distinct types of entities: points <italic>P</italic>, <italic>Q</italic>, <italic>R</italic> … and vectors <italic>u</italic>, <italic>v</italic>, <italic>w</italic> … Generally, the operattions of vector algebra —addition, subtraction, scalar multiplication, dot product, and cross product—are intrinsically defined only for vectors, not for points. Yet illicit expressions containing terms like <italic>P</italic> + <italic>Q</italic>, <italic>cP</italic>, <italic>P</italic> X <italic>Q</italic>, etc. often appear in graphics textbooks, papers, and programs. In this paper we justify the use of such illicit expressions, and we we give criteria for recognizing when such an expression is truly legitimate. In particular we show that an algebraic expression <italic>E</italic>(<italic>P</italic><sub scrpt>1</subscrpt>, …, <italic>Pn</italic>) is legitimate if and onl y if <italic>E</italic>(<italic>v</italic><subscrpt>1</subscrpt> + <italic> w</italic>, …<italic>v<subscrpt>n</subscrpt></italic> + <italic>w</italic>) = <italic>E</italic>(<italic>v<subscrpt>1</subscrpt></italic>, …, <ita lic>v<subscrpt>n</subscrpt></italic>) + <italic>kw</italic>, <italic>k</italic> + 0, 1. We also derive many useful examples of such an expression.