4.1: Addition and Scalar Multiplication in Rⁿ
( \newcommand{\kernel}{\mathrm{null}\,}\)
A simple but important property of n-vectors is that we can add n-vectors and multiply n-vectors by a scalar:
Definition
Given two n-vectors a and b whose components are given by
a=(a1⋮an) and b=(b1⋮bn)
a+b:=(a1+b1⋮an+bn).
Given a scalar λ, the scalar multiple
$$\lambda a := (λa1⋮λan)
$$a+b= (5555)
A special vector is the zero vector. All of its components are zero:
$$0=(0⋮0)
In Euclidean geometry---the study of Rn 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.