Page 84 - 35Linear Algebra

84 Vectors in Space, n-Vectors

4.1 Addition and Scalar Multiplication in R n

A simple but important property of n-vectors is that we can add two n-vectors
together and multiply one n-vector by a scalar:

Definition Given two n-vectors a and b whose components are given by
   
a 1 b 1
. .
 .   . 
a =  .  and b =  . 
a n b n
their sum is
 
1
a + b 1
.
 . .  .

a + b := 
n
a + b n
Given a scalar λ, the scalar multiple
 
λa 1
.
λa :=  .  .
 . 
λa n
Example 44 Let
   
1 4
2 3
   
a =   and b =   .
3 2
   
4 1
Then, for example,
   
5 −5
5
   0 
a + b =   and 3a − 2b =   .
5 5
   
5 10
A special vector is the zero vector. All of its components are zero:
 
0
.
0 =  .  =: 0 n .
.
0
n
In Euclidean geometry—the study of R with lengths and angles defined
as in section 4.3 —n-vectors are used to label points P and the zero vector
labels the origin O. In this sense, the zero vector is the only one with zero
magnitude, and the only one which points in no particular direction.

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