Advanced Calculus fi..

Chapter 1 Vectors and Matrices 3Figure 1.2 Vector v in terms of i, j, k.Figure 1.3Definition of dot product.More generally, for v = PI 6, where PI is (XI, yl, 2,) and P2 is (x2, y2, z2), one has* *v = OP2 - OP1 = (x2- xl)i + ... and the distance between PI and P2 isThe vector v can have 0 length, in which case v = 3 only when P coincideswith 0. We then writeand call v the zero vector.The vector v is completely specified by its components v,, v,, v,. It is oftenconvenient to writeinstead of Eq. (1.3). Thus we think of a vector in space as an ordered triple ofnumbers. Later we shall consider such triples as matrices (row vectors or columnvectors).The dot product (or inner product) of two vectors v, w in space is the numberwhere 0 = ~ (v, w), chosen between 0 and n inclusive (see Fig. 1.3). When v or wis 0, the angle 6 is indeterminate, and v . w is taken to be 0. We also have v . w = 0when v, w are orthogonal (perpendicular) vectors, v I w. We agree to say that the 0vector is orthogonal to all vectors (and parallel to all vectors). With this conventi