Scalar Multiplication of Vectors Using Components
Given a single vector v = (v_{1}, v_{2}) in the Euclidean plane, and a scalar a
(which is a real number), the multiplication of the vector by the scalar is
defined as:
Similarly, for a 3dimensional vector v = (v_{1}, v_{2}, v_{3}) and a scalar a, the
formula for scalar multiplication is:
av = (av_{1}, av_{2}, av_{3}) 

So what we are doing when we multiply a vector by a scalar a is obtaining a
new vector (of the same dimension) by multiplying each component of the
original vector by a.
Unit Vectors
For 3dimensional vectors, it is often customary to define unit vectors
pointing in the x, y, and z directions. These vectors are usually denoted
by the letters i, j, and k, respectively, and all have
length 1. Thus, i = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1).
This enables us to write a vector as a sum in the following way:
(a, b, c)  =  a(1, 0, 0) + b(0, 1, 0) + c(0, 0, 1) 

 =  ai + bj + ck 

Vector Subtraction
Subtraction for vectors (as with ordinary numbers) is not a new operation.
If you want to perform the vector subtraction u  v, you simply use the rules
for vector addition and scalar multiplication: u  v = u + ( 1)v.
In the next section, we will see
how these rules for addition and scalar multiplication of vectors can be
understood in a geometric way. We will find, for instance, that vector addition
can be done graphically (i.e. without even knowing the components of the vectors
involved), and that scalar multiplication of a vector amounts to a change in the
vector's magnitude, but does not alter its direction.