Multivariate Calculus - Bruce E. Shapiro

Lecture 2
Vectors in 3D
Properties of Vectors
Definition 2.1 A displacement vector v from point A to point B is an arrow
pointing from A to B, and is denoted as
v =
⃗ AB (2.1)
In general, we will use the same notation (e.g., a boldface letter such as v) to
denote a vector that we use to describe a point (such as A) as well as a matrix. In
most cases (but not always) we will use upper case letters for points (or matrices)
and lower case letters for vectors. Whenever there is any ambiguity we will write
a small arrow over the symbol for the vector, as in ⃗v, which means the same thing
as v. A small arrow over a pair of points written next to each other, as in AB ⃗ is
used to denote the displacement vector pointing from A to B. If v is a vector then
v denotes its magnitude:
Definition 2.2 The length or magnitude of a vector v is the distance measured
from one end point to the other, and is denoted by the following equivalent notations:
v = |v| = ‖v‖ (2.2)
In print the notation v is more common for a vector; in handwritten documents
(and some textbooks) it is usual to write ⃗v for a vector.
Figure 2.1: Concept of a vector as the difference between two points.
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