Vector addition

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Suppose that the vector displacement
${\bf r}$ of some point $R$ from the origin $O$ is specified as follows:

\begin{displaymath}
{\bf r} = {\bf r}_1 + {\bf r}_2.
\end{displaymath}

(31)


Figure 12 illustrates how this expression is interpreted diagrammatically: in order to
get from point $O$ to point $R$, we first move from point $O$ to point $S$ along vector
${\bf r}_1$, and we then move from point $S$ to point $R$ along vector ${\bf r}_2$. The
net result is the same as if we had moved from point $O$ directly to point $R$ along
vector ${\bf r}$. Vector ${\bf r}$ is termed the resultant of adding vectors
${\bf r}_1$ and ${\bf r}_2$.

Figure 12:
Vector addition
\begin{figure}
\epsfysize =2in
\centerline{\epsffile{vadd.eps}}
\end{figure}

Note that we have two ways of specifying the vector displacement of point $S$ from
the origin: we can either write ${\bf r}_1$ or
${\bf r} - {\bf r}_2$. The
expression
${\bf r} - {\bf r}_2$ is interpreted as follows: starting at the origin,
move along vector ${\bf r}$ in the direction of the arrow, then move along
vector ${\bf r}_2$ in the opposite direction to the arrow. In other words,
a minus sign in front of a vector indicates that we should move along that vector in
the opposite direction to its arrow.

Suppose that the components of vectors ${\bf r}_1$ and ${\bf r}_2$ are

$(x_1, y_1, z_1)$ and
$(x_2, y_2, z_2)$, respectively. As is easily demonstrated,
the components $(x,y,z)$ of the
resultant vector
${\bf r}= {\bf r}_1 + {\bf r}_2$ are

$\displaystyle x$
$\textstyle =$
$\displaystyle x_1+x_2,$

(32)
$\displaystyle y$
$\textstyle =$
$\displaystyle y_1 + y_2,$

(33)
$\displaystyle z$
$\textstyle =$
$\displaystyle z_1 + z_2.$

(34)

In other words, the components of the sum of two vectors are simply the algebraic
sums of the components of the individual vectors.

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Next: Vector magnitude
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Previous: Vector displacement

Richard Fitzpatrick
2006-02-02