Given a single vector v = (v1, v2) in the Euclidean plane, and a scalar a (which is a real number), the multiplication of the vector by the scalar is defined as:
|av = (av1, av2)|
Similarly, for a 3-dimensional vector v = (v1, v2, v3) and a scalar a, the formula for scalar multiplication is:
|av = (av1, av2, av3)|
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.
For 3-dimensional 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|
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.