Projects - Cengage Learning

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0
0
0
0
C-76 Appendix C
Exercise 4 Draw a nonzero vector v and its opposite, v. Then draw the vector
v (v) to show
v (v) O (3)
Now we want to define kv, where k is a negative real number.
Exercise 5 Draw a nonzero vector v. Then draw a vector with the opposite direction
of v and with magnitude twice that of v. On the basis of our definition
of v, show that we can construct the desired vector by adding two copies of
v, that is, (v) (v).
Exercises 5 and 3 suggest it is natural to denote this new vector by the symbol
2v, that is, (v) (v) 2v. The 2 indicates the vector 2v has direction
opposite that of v and magnitude 2 2 times that of v. This leads us
to define, for a negative real number k, the vector kv as the vector with direction
opposite that of v and magnitude k times that of v. In symbols, kv k 00 v .
Note that k is the absolute value of the number k and v is the magnitude of
the vector v. Note also that for k positive we have kv k v k v , too.
This leads us to a definition.
Definition
Scalar Multiplication
Given a nonzero vector v and a nonzero real number k, the vector kv is the vector
of magnitude kv k 00 v| whose direction is the same as that of v if k is
positive and opposite that of v if k is negative. If either v O or k 0, then
we define kv to be the zero vector, that is, we define kO O and 0v O.
“Multiplying” a vector by a real number is called scalar multiplication.
v
u-v
Exercise 6 For any vector v, explain why 1v v and 1v v.
Now we want to define subtraction of vectors. In arithmetic we define the
difference of two numbers to be the number you add to the number subtracted
to get the number you started with. For example, 8 5 3, since 5 3 8,
that is, 3 [or (8 5)] is what you add to 5 to get 8. Now let u and v be vectors
and consider a new vector called the difference of u and v, denoted by u v.
By analogy to subtraction of numbers, we define the difference u v to be the
vector that you add to v to get u. See Figure A.
u
Figure A
u v is the vector that
you add to v to get u.
Exercise 7 Draw two vectors u and v with different directions and with v not
equal to the opposite of u. Show
u v u (v) (4)
So we can think of subtracting as adding the opposite.