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Friday, 8 August 2014

CHAPTER 3-Subtraction of Vectors

Consider two vectors \vec a and \vec b; we wish to find \vec c such that
\vec c = \vec a - \vec b
We can slightly modify this relation and write it as
\vec c = \vec a + \left( { - \vec b} \right)
and thus subtraction can be treated as addition. To do this, we first reverse the vector \vec b to obtain  - \,\vec b and then use the triangle / parallelogram law of addition to add the vector \vec a and ( - \vec b):
Joining the tip of \vec b to the tip of \vec a (if \vec a and \vec b are co-initial) also gives us \vec a - \vec b.
Note that from the triangle law, it follows that for three vectors \vec a,\vec b and \vec crepresenting the sides of a triangle as shown,
We must have
\vec a + \vec b + \vec c = \vec 0
In fact, for the vectors {\vec a_i},\,\,i = 1,2\ldots n, representing the sides of an n-sided polygon as shown,
we must have
{\vec a_1} + {\vec a_2} + \ldots+ {\vec a_n} = \vec 0
since the net effect of all vectors is to bring us back from where we started, and thus our net displacement is the zero vector.
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