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

Section 12.2 Vectors 803
y
a
(a¡, a)
aj
If a − ka 1 , a 2 , a 3 l, then we can write
a − ka 1 , a 2 , a 3 l − ka 1 , 0, 0 l 1 k0, a 2 , 0 l 1 k0, 0, a 3 l
− a 1 k1, 0, 0 l 1 a 2 k0, 1, 0 l 1 a 3 k0, 0, 1 l
0
a¡i
x
2 a − a 1 i 1 a 2 j 1 a 3 k
(a) a=a¡i+a j
Thus any vector in V 3 can be expressed in terms of i, j, and k. For instance,
z
a
(a¡, a, a£)
Similarly, in two dimensions, we can write
k1, 22, 6l − i 2 2j 1 6k
3 a − ka 1 , a 2 l − a 1 i 1 a 2 j
a¡i
a£k
x
aj
(b) a=a¡i+a j+a£k
FIGURE 18
y
See Figure 18 for the geometric interpretation of Equations 3 and 2 and compare with
Figure 17.
ExamplE 5 If a − i 1 2j 2 3k and b − 4i 1 7 k, express the vector 2a 1 3b in
terms of i, j, and k.
SOLUtion Using Properties 1, 2, 5, 6, and 7 of vectors, we have
2a 1 3b − 2si 1 2j 2 3kd 1 3s4i 1 7kd
Gibbs
Josiah Willard Gibbs (1839–1903), a
professor of mathematical physics
at Yale College, published the first
book on vectors, Vector Analysis, in
1881. More complicated objects,
called quaternions, had earlier been
invented by Hamilton as mathematical
tools for describing space, but
they weren’t easy for scientists to use.
Quaternions have a scalar part and
a vector part. Gibb’s idea was to use
the vector part separately. Maxwell
and Heaviside had similar ideas, but
Gibb’s approach has proved to be the
most convenient way to study space.
− 2i 1 4j 2 6k 1 12i 1 21k − 14i 1 4j 1 15k
A unit vector is a vector whose length is 1. For instance, i, j, and k are all unit vectors.
In general, if a ± 0, then the unit vector that has the same direction as a is
4 u − 1
| a | a − a
| a |
In order to verify this, we let c − 1y| a | . Then u − ca and c is a positive scalar, so u has
the same direction as a. Also
| u | − | ca | − | c | | a | − 1
| a | | a | − 1
ExamplE 6 Find the unit vector in the direction of the vector 2i 2 j 2 2k.
■
SOLUtion The given vector has length
| 2i 2 j 2 2k | − s2 2 1 s21d 2 1 s22d 2 − s9 − 3
so, by Equation 4, the unit vector with the same direction is
Applications
1
3 s2i 2 j 2 2kd − 2 3 i 2 1 3 j 2 2 3 k ■
Vectors are useful in many aspects of physics and engineering. In Chapter 13 we will see
how they describe the velocity and acceleration of objects moving in space. Here we look
at forces.
A force is represented by a vector because it has both a magnitude (measured in
pounds or newtons) and a direction. If several forces are acting on an object, the resultant
force experienced by the object is the vector sum of these forces.
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