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Suppose that the displacements
and
represent the vectors and , respectivelysee Figure A.97. It can be seen that the result
of combining these two displacements is to give the net displacement
. Hence,
if
represents the vector then we can write

(1261) 
This defines vector addition.
By completing the parallelogram , we can also see that

(1262) 
However,
has the same length and direction as
,
and, thus, represents the same vector, . Likewise,
and
both represent the vector . Thus, the above equation is equivalent to

(1263) 
We conclude that the addition of vectors is commutative. It can also
be shown that the associative law holds: i.e.,

(1264) 
The null vector, , is represented by a displacement of zero length and arbitrary direction.
Since the result of combining such a displacement with a finite length displacement is the same
as the latter displacement by itself, it follows that

(1265) 
where is a general vector.
The negative of is defined as that vector which has the same magnitude, but acts in the opposite direction, and is denoted .
The sum of and is thus
the null vector: i.e.,

(1266) 
We can also define the difference of two vectors, and , as

(1267) 
This definition of vector subtraction is illustrated in Figure A.98.
Figure A.98:
Vector subtraction.

If is a scalar then the expression denotes a vector whose direction is the same
as , and whose magnitude
is times that of . (This definition becomes obvious when is an integer.)
If is negative then, since
, it follows
that is a vector whose magnitude is times that of , and whose
direction is opposite to . These definitions imply that if and are
two scalars then
Next: Cartesian Components of a
Up: Vector Algebra and Vector
Previous: Scalars and Vectors
Richard Fitzpatrick
20110331