Multivariate Calculus - Bruce E. Shapiro

10 LECTURE 2. VECTORS IN 3D
If the point A = (x a , y a , z a ) and the point B = (x b , y b , z b ) then we define the
components of the vector v = AB ⃗ as v = (v x , v y , v z ), where
v = (v x , v y , v z ) = (x b − x a , y b − y a , z b − z a ) (2.3)
We observe that the difference of two points is a vector, and write this as
v =
⃗ AB = B − A (2.4)
There is no equivalent concept of the sum of two points, although we will see that
it is possible to add two vectors.
The text uses angle brackets 〈〉 to denote a vector in terms of its components:
〈a, b, c〉 = (a, b, c) (2.5)
The use of parenthesis is more common, although angle brackets are used whenever
there is some possibility of confusion between vectors and points.
Theorem 2.1 The magnitude of a vector v = (v x , v y , v z ) is given by
‖v‖ =
√
v 2 x + v 2 y + v 2 z (2.6)
Proof. According to the distance formula, the distance from A to B is
‖v‖ = √ √
(x b − x a ) 2 + (y b − y a ) 2 + (z b − z a ) 2 = vx 2 + vy 2 + vz.
2
Definition 2.3 Two vectors v and w are said to be equal if they have the same
magnitude and direction i.e., they have the same length and are parallel, and we
write v = w
Theorem 2.2 Two vectors v = (v x , v y , v z ) and w = w x , w y , w z are equal if and
only if their components are equal, namely the following three conditions all hold:
v x = w x , v y = w y , v z = w z (2.7)
Definition 2.4 The zero vector, denoted by 0, is a vector with zero magnitude
(and undefined direction).
Operations on Vectors
Definition 2.5 Scalar Multiplication or multiplication of a vector by a
scalar is defined as follows. Suppose that a ∈ R is any real number and v =
v x , v y , v z ∈ R 3 is a vector. Then
av = a(v x , v y , v z ) = (av x , av y , av z ) (2.8)
Revised December 6, 2006. Math 250, Fall 2006