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 $ \underline{w}\isdef \underline{x}+\underline{y}$, then we have $ \underline{w}_n = x_n+y_n$ for $ n=0,1,2,\ldots,N-1$. We could also write $ \underline{w}(n) = x(n)+y(n)$ for $ n=0,1,2,\ldots,N-1$ if preferred.

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$, $ \underline{x}=(2,3)$, and $ \underline{y}=(4,1)$.

Figure 5.2: Geometric interpretation of a length 2 vector sum.
\includegraphics[scale=0.7]{eps/vecsum}

Also shown are the lighter construction lines which complete the parallelogram started by $ \underline{x}$ and $ \underline{y}$, indicating where the endpoint of the sum $ \underline{x}+\underline{y}$ lies. Since it is a parallelogram, the two construction lines are congruent to the vectors $ \underline{x}$ and $ \underline{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 5.3: Vector sum, translating one vector to the tip of the other.
\includegraphics[scale=0.7]{eps/vecsumr}

In the figure, $ \underline{x}$ was translated to the tip of $ \underline{y}$. This depicts $ y+x$, since ``$ x$ picks up where $ y$ leaves off.'' It is equally valid to translate $ \underline{y}$ to the tip of $ \underline{x}$, because vector addition is commutative, i.e., $ \underline{x}+\underline{y}$ = $ \underline{y}+\underline{x}$.


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