## Vector Addition

Given two vectors in , say

the *vector sum* is defined by
*elementwise* addition. If we denote the sum by
,
then we have
for
. We could also
write
for
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 , , and .

Also shown are the lighter construction lines which complete the
parallelogram started by
and
, indicating where the endpoint of the
sum
lies. Since it is a parallelogram, the two construction lines
are congruent to the vectors
and
. 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.

In the figure,
was translated to the tip of
.
This depicts , since `` picks up where leaves off.''
It is equally valid
to translate
to the tip of
, because vector addition is *commutative*, *i.e.*,
=
.

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Vector Subtraction

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