James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

802 Chapter 12 Vectors and the Geometry of Space
Vectors in n dimensions are used to list
various quantities in an organized way.
For instance, the components of a sixdimensional
vector
p − k p 1, p 2, p 3, p 4, p 5, p 6 l
might represent the prices of six different
ingredients required to make a
particular product. Four-dimensional
vectors k x, y, z, t l are used in relativity
theory, where the first three components
specify a position in space and the
fourth represents time.
We denote by V 2 the set of all two-dimensional vectors and by V 3 the set of all threedimensional
vectors. More generally, we will later need to consider the set V n of all
n-dimensional vectors. An n-dimensional vector is an ordered n-tuple:
a − k a 1 , a 2 , . . . , a n l
where a 1 , a 2 , . . . , a n are real numbers that are called the components of a. Addition and
scalar multiplication are defined in terms of components just as for the cases n − 2 and
n − 3.
Properties of Vectors If a, b, and c are vectors in V n and c and d are scalars, then
1. a 1 b − b 1 a 2. a 1 sb 1 cd − sa 1 bd 1 c
3. a 1 0 − a 4. a 1 s2ad − 0
5. csa 1 bd − ca 1 cb 6. sc 1 dda − ca 1 da
7. scdda − csdad 8. 1a − a
Q
c
These eight properties of vectors can be readily verified either geometrically or algebraically.
For instance, Property 1 can be seen from Figure 4 (it’s equivalent to the Parallelogram
Law) or as follows for the case n − 2:
a 1 b − ka 1 , a 2 l 1 kb 1 , b 2 l − ka 1 1 b 1 , a 2 1 b 2 l
− kb 1 1 a 1 , b 2 1 a 2 l − kb 1 , b 2 l 1 ka 1 , a 2 l
− b 1 a
(a+b)+c
=a+(b+c)
a+b
b+c
b
We can see why Property 2 (the associative law) is true by looking at Figure 16 and
applying the Triangle Law several times: the vector PQ l is obtained either by first constructing
a 1 b and then adding c or by adding a to the vector b 1 c.
Three vectors in V 3 play a special role. Let
P
a
FIGURE 16
i − k1, 0, 0l j − k0, 1, 0l k − k0, 0, 1l
These vectors i, j, and k are called the standard basis vectors. They have length 1 and
point in the directions of the positive x-, y-, and z-axes. Similarly, in two dimensions we
define i − k1, 0l and j − k 0, 1l. (See Figure 17.)
y
z
(0, 1)
FIGURE 17
Standard basis vectors in V 2 and V 3
j
0
(a)
i
(1, 0)
x
x
k
i
(b)
j
y
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