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# Vector Algebra

Suppose that the displacements and represent the vectors and , respectively--see 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.

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

 (1268) (1269) (1270)

Next: Cartesian Components of a Up: Vector Algebra and Vector Previous: Scalars and Vectors
Richard Fitzpatrick 2011-03-31