# Vector Addition

We now proceed to define addition of vectors in $R^2$, and multiplication of vectors in $R^2$by real numbers. In this context, we interpret $R^2$ as a set of vectors represented by arrowswith tail at the origin.

Given two vectors $a=(a_1,a_2)$ and $b=(b_1,b_2)$ in $R^2$, we use $a+b$ to denote the vector $(a_1+b_1,a_2+b_2)$ in $R^2$ obtained by adding the components separately. We call $a+b$ the sum of $a$ and $b$ obtained through vector addition. Thus if  $a=(a_1,a_2)$  and  $b=(b_1,b_2)$ are given vectors in $R^2$, then

• $(a_1,a_2)+(b_1,b_2)=(a_1+b_1,a_2+b_2)$,

which says that vector addition is carried out by adding components separately. We note that $a+b=b+a$ since $a_1+b_1=b_1+a_1$ and $a_2+b_2=b_2+a_2$. We say that $0=(0,0)$ is the zero vector since $a+0=0+a=a$ for any vector $a$. Note the difference between the vector zero and its two zero components, which are usually scalars

Example: We have $(2,5)+(7,1)=(9,6)$ and $(2.1,5.3)+(7.6,1.9)$ $=(9.7,7.2)$.

# Vector Addition and the Parallelogram Law

We may represent vector addition geometrically using the Parallelogram Law as follows. The vector $a+b$ corresponds to the arrow along the diagonal in the parallelogram with two sides formed by the arrows $a$ and $b$:

This follows by noting that the coordinates of the head of $a+b$ is obtained by adding the coordinates of the points $a$ and $b$ separately, as just illustrated.

This definition of vector addition implies that we may reach the point $(a_1+b_1,a_2+b_2)$ by walking along arrows in two different ways.

First, we simply follow the arrow $(a_1+b_1,a_2+b_2)$ to its head, corresponding to walking along the diagonal of the parallelogram formed by $a$ and $b$.

Secondly, we could follow the arrow $a$ from the origin to its head at the point $(a_1,a_2)$ and then continue to the head of the arrow $\bar b$ parallel to $b$ and of equal length as $b$ with tail at $(a_1,a_2)$.

Alternative, we may follow the arrow $b$ from the origin to its head at the point $(b_1,b_2)$ and then continue to the head of the arrow $\bar a$ parallel to $a$ and of equal length as $a$ with tail at $(b_1,b_2)$.

The three different routes to the point latex $(a_1+b_1,a_2+b_2)$ are displayed in the above figure.

We sum up in the following theorem:

Theorem: Adding two vectors $a=(a_1,a_2)$ and $b=(b_1,b_2)$  in $R^2$ to get the sum $a+b=(a_1+b_1,a_2+b_2)$  corresponds to adding the arrows $a$ and $b$ using the Parallelogram Law.

In particular, we can write a vector as the sum of its components in the coordinate directions as follows:

• $(a_1,a_2)=(a_1,0)+(0,a_2)$,
as illustrated in the following figure:
A vector $a$ represented as the sum of two vectors parallel with the coordinate axes.