Comparing scalars and vectors

Scalars are regular numbers, such as 7, 82, and 93,454. They only have a magnitude and are used to represent time, speed, distance, length, mass, work, power, area, volume, and so on.

Vectors, on the other hand, have magnitude and direction in many dimensions. We use vectors to represent velocity, acceleration, displacement, force, and momentum. We write vectors in bold—such as a instead of a—and they are usually an array of multiple numbers, with each number in this array being an element of the vector. 

We denote this as follows:

Here, shows the vector is in n-dimensional real space, which results from taking the Cartesian product of  n times;  shows each element is a real number; i is the position of each element; and, finally,  is a natural number, telling us how many elements are in the vector. 

As with regular numbers, you can add and subtract vectors. However, there are some limitations. 

Let's take the vector we saw earlier (x) and add it with another vector (y), both of which are in , so that the following applies:

However, we cannot add vectors with vectors that do not have the same dimension or scalars.

Note that when  in , we reduce to 2-dimensions (for example, the surface of a sheet of paper), and when n = 3, we reduce to 3-dimensions (the real world). 

We can, however, multiply scalars with vectors. Let λ be an arbitrary scalar, which we will multiply with the vector , so that the following applies:

As we can see, λ gets multiplied by each x_i in the vector. The result of this operation is that the vector gets scaled by the value of the scalar.

For example, let , and . Then, we have the following:

While this works fine for multiplying by a whole number, it doesn't help when working with fractions, but you should be able to guess how it works. Let's see an example.

Let , and . Then, we have the following:

There is a very special vector that we can get by multiplying any vector by the scalar, 0. We denote this as 0 and call it the zero vector (a vector containing only zeros).