Addition and subtraction of vectors

7.2 ADDITION AND SUBTRACTION OF VECTORS
a
b+a
b
a+b
b
a
Figure 7.1 Addition of two vectors showing the commutation relation. We
make no distinction between an arrowhead at the end of the line and one
along the line’s length, but rather use that which gives the clearer diagram.
end of the line and one along the line’s length but, rather, use that which gives the
clearer diagram. Furthermore, even though we are considering three-dimensional
vectors, we have to draw them in the plane of the paper. It should not be assumed
that vectors drawn thus are coplanar, unless this is explicitly stated.
7.2 Addition and subtraction of vectors
The resultant or vector sum of two displacement vectors is the displacement vector
that results from performing first one and then the other displacement, as shown
in figure 7.1; this process is known as vector addition. However, the principle
of addition has physical meaning for vector quantities other than displacements;
for example, if two forces act on the same body then the resultant force acting
on the body is the vector sum of the two. The addition of vectors only makes
physical sense if they are of a like kind, for example if they are both forces
acting in three dimensions. It may be seen from figure 7.1 that vector addition is
commutative, i.e.
a + b = b + a.
(7.1)
The generalisation of this procedure to the addition of three (or more) vectors is
clear and leads to the associativity property of addition (see figure 7.2), e.g.
a + (b + c) = (a + b) + c.
(7.2)
Thus, it is immaterial in what order any number of vectors are added.
The subtraction of two vectors is very similar to their addition (see figure 7.3),
that is,
a − b = a + (−b)
where −b is a vector of equal magnitude but exactly opposite direction to vector b.
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