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Appendix C C-75
PROJECT
Vector Algebra Using Vector Geometry
In the previous section we defined vectors and vector addition geometrically
and showed that vector addition is commutative, that is, if u and v are vectors
then u v v u. What happens if we want to add three vectors? Let u, v,
and w be vectors. How should we define the sum u v w? Addition was
defined for two vectors at a time. That leads to two possible “groupings” of the
three terms. We might add u and v first, then add w to their sum. In symbols we
write (u v) w, where the parentheses are grouping symbols, as in algebra,
indicating what operation is done first. Alternatively we might add v and w
first, then add their sum to u. In symbols we write u (v w). Of course, we
would like (u v) w to equal u (v w). If this is true, then we can write
the sum u v w without ambiguity.
Exercise 1 Draw three distinct (unequal) vectors u, v, and w. Use the definition
of vector addition to compute (u v) w and u (v w). Convince
yourself that the resulting sums are equal. So we have
(u v) w u (v w) (1)
the associative property of vector addition.
Now we want to define a vector with the property that when we add it to any
vector v, it leaves v unchanged. This new vector is called the zero vector, denoted
O. So
v O v (2)
According to equation (2), the zero vector is a vector with no direction and
whose magnitude is zero. It can be shown that the zero vector is unique, that is,
O is the only vector with the property that v O v for all vectors v.
Exercise 2 Explain why the zero vector must have no direction and have
magnitude zero.
The next exercise provides a motivation for a definition of the multiplication
of a vector by a positive real number.
Exercise 3 Draw a nonzero vector v. Then draw the vector v v. What are
the direction and magnitude of v v?
On the basis of Exercise 3 it is natural to call the vector v v, 2v, that is,
v v 2v. Similarly, for any positive real number k the vector kv is defined
to be the vector with the same direction as v and whose magnitude is k times
the magnitude of v. In symbols, 0 kv k v .
Now start with a nonzero vector v. Consider a new vector with the “opposite”
direction of v and the same magnitude as v. By opposite direction we
mean that this new vector is represented by an arrow whose initial point is the
terminal point of v and whose terminal point is the initial point of v. This vector
is called the opposite of v, denoted v. Clearly, given v, the vector v is
unique.