We now proceed to define addition of vectors in , and multiplication of vectors in by real numbers. In this context, we interpret as a set of vectors represented by arrowswith tail at the origin.
Given two vectors and in , we use to denote the vector in obtained by adding the components separately. We call the sum of and obtained through vector addition. Thus if and are given vectors in , then
which says that vector addition is carried out by adding components separately. We note that since and . We say that is the zero vector since for any vector . Note the difference between the vector zero and its two zero components, which are usually scalars
Example: We have and .
Vector Addition and the Parallelogram Law
We may represent vector addition geometrically using the Parallelogram Law as follows. The vector corresponds to the arrow along the diagonal in the parallelogram with two sides formed by the arrows and :
This follows by noting that the coordinates of the head of is obtained by adding the coordinates of the points and separately, as just illustrated.
This definition of vector addition implies that we may reach the point by walking along arrows in two different ways.
First, we simply follow the arrow to its head, corresponding to walking along the diagonal of the parallelogram formed by and .
Secondly, we could follow the arrow from the origin to its head at the point and then continue to the head of the arrow parallel to $b$ and of equal length as with tail at .
Alternative, we may follow the arrow from the origin to its head at the point and then continue to the head of the arrow parallel to and of equal length as with tail at .
The three different routes to the point latex are displayed in the above figure.
We sum up in the following theorem:
Theorem: Adding two vectors and in to get the sum corresponds to adding the arrows and using the Parallelogram Law.
In particular, we can write a vector as the sum of its components in the coordinate directions as follows: