Multivariate Calculus - Bruce E. Shapiro

LECTURE 2. VECTORS IN 3D 11
Figure 2.2: Scalar multiplication.
Scalar multiplication changes the length of a vector, as illustrated in figure 2.2. As
a consequence of definitions 2.3 and 2.4, we also conclude that
0 = (0, 0, 0) (2.9)
Definition 2.6 Vector additive inverse or the negative of a vector. Suppose
that v = (v x , v y , v z ). Then
−v = (−1)v = (−v x , −v y , −v z ) (2.10)
Figure 2.3: Illustration of vector addition. Top: the three vectors u, v, and w =
u + v. Bottom: vector addition, and illustration of commutative law for vector
addition: u + v = v + u.
Vector addition, illustrated in figure 2.3 proceeds as follows: Join the two
vectors head to tail as show in the figure to the right and then draw the arrow from
the tail to the head. Addition is commutative: the order in which the vectors are
added does not matter. A vector sum can be calculated component-by-component;
suppose that v = (v x , v y , v z ) and u = (u x , u y , u z ). Then
u + v = (u x , u y , u z ) + (v x , v y , v z ) = (u x + v x , u y + v y , u z + v z ) (2.11)
Vector subtraction is similar, and is induced by the idea that we want
v − u = v + −u (2.12)
Math 250, Fall 2006 Revised December 6, 2006.