Multivariate Calculus - Bruce E. Shapiro

12 LECTURE 2. VECTORS IN 3D
To obtain u - v, start (1) by placing the second vector at the head of the first and
then (2) reflect it across its own tail to find −u. Finally, (3) draw the arrow from
the tail of u to the head of the reflection of v. Observe that this process is not
reversible: u - v is not the same as v - u. In terms of components,
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 ) = −(v − u) (2.13)
Figure 2.4: Vector subtraction, demonstrating construction of v −u (left) and v −u
(right). Observe that v − u = −(v − u)
Theorem 2.3 Properties of Vector Addition & Scalar Multiplication. Let
v, u, and w be vectors, and a, b ∈ R be real numbers. Then the following properties
hold:
1. Vector addition commutes:
2. Vector addition is associative:
v + u = u + v (2.14)
(u + v) + w = u + (v + w) (2.15)
3. Scalar multiplication is distributive across vector addition:
4. Identity for scalar multiplication
5. Properties of the zero vector
(a + b)v = av + bv (2.16)
a(v + w) = av + aw (2.17)
1v = v1 = v (2.18)
0v = 0 (2.19)
0 + v = v + 0 = v (2.20)
Revised December 6, 2006. Math 250, Fall 2006