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# Vector Algebra

Suppose that the displacements and , shown in Figure A.2, represent the vectors and , respectively. 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 (A.1)

This defines vector addition. By completing the parallelogram , we can also see that (A.2)

However, has the same length and direction as , and, thus, represents the same vector, . Likewise, and both represent the vector . Thus, the previous equation is equivalent to (A.3)

We conclude that the addition of vectors is commutative. It can also be shown that the associative law holds: that is, (A.4)

The null vector, , is represented by a displacement of zero length and arbitrary direction. Because the result of combining such a displacement with a finite length displacement is the same as the latter displacement by itself, it follows that (A.5)

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: that is, (A.6)

We can also define the difference of two vectors, and , as (A.7)

This definition of vector subtraction is illustrated in Figure A.3. 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, because , 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  (A.8)  (A.9)  (A.10)   Next: Cartesian Components of a Up: Vectors and Vector Fields Previous: Scalars and Vectors
Richard Fitzpatrick 2016-03-31