Vector Addition

Given two vectors in ${\bf R}^N$, say

$$\begin{eqnarray*} \underline{x}&\isdef & (x_0,x_1,\ldots,x_{N-1})\\ \underline{y}&\isdef & (y_0,y_1,\ldots,y_{N-1}), \end{eqnarray*}$$

the vector sum is defined by elementwise addition. If we denote the sum by $w\isdef x+y$, then we have $w(n) = x(n)+y(n)$ for $n=0,1,2,\ldots,N-1$.

The vector diagram for the sum of two vectors can be found using the parallelogram rule, as shown in Fig. 5.2 for $N=2$, $x=(2, 3)$, and $y=(4,1)$.

Figure: Geometric interpretation of a length 2 vector sum.

Also shown are the lighter construction lines which complete the parallelogram started by $x$ and $y$, indicating where the endpoint of the sum $x+y$ lies. Since it is a parallelogram, the two construction lines are congruent to the vectors $x$ and $y$. As a result, the vector sum is often expressed as a triangle by translating the origin of one member of the sum to the tip of the other, as shown in Fig. 5.3.

Figure: Vector sum, translating one vector to the tip of the other.

In the figure, $x$ was translated to the tip of $y$. It is equally valid to translate $y$ to the tip of $x$, because vector addition is commutative, i.e., $x+y$ = $y+x$.