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ESSENTIAL CALCULUSESSENTIAL CALCULUS
CH10 Vectors and the CH10 Vectors and the geometry of spacegeometry of space
In this Chapter:In this Chapter:
10.1 Three-Dimensional Coordinate Systems
10.2 Vectors
10.3 The Dot Product
10.4 The Cross Product
10.5 Equations of Lines and Planes
10.6 Cylinders and Quadric Surfaces
10.7 Vector Functions and Space Curves
10.8 Arc Length and Curvature
10.9 Motion in Space: Velocity and Acceleration
Review
Chapter 10, 10.1, P519
Chapter 10, 10.1, P519
Chapter 10, 10.1, P519
Chapter 10, 10.1, P519
Chapter 10, 10.1, P520
Chapter 10, 10.1, P521
Chapter 10, 10.1, P521
DISTANCE FORMULA IN THREE IMENSIONS The distance │P1P2│between the points P1(x1,y1,z1) and P2(x2,y2,z2) is
212
212
21221 )()()( zzyyxxPP
Chapter 10, 10.1, P522
EQUATION OF A SPHERE An equation of a sphere with center C( h, k, l) and radius r is
In particular, if the center is the origin O , then an equation of the sphere is
2222 )()()( rlzkyhx
2222 rzyx
Chapter 10, 10.2, P524
Chapter 10, 10.2, P524
Chapter 10, 10.2, P524
The term vector is used by scientists to indicate a quantity (such as displacement or velocity or force) that has both magnitude and direction. A vector is often represented by an arrow or a directed line segment.
We denote a vector by printing a letter in boldface (v) or by putting an arrow above the letter (v).
Chapter 10, 10.2, P524
displacement vector v , shown in Figure 1, has initial point A (the tail) and terminal point B (the tip) and we indicate this by writing v=AB. Notice that the vector u=CD has the same length and the same direction as v even though it is in a different position. We say that u and v are equivalent (or equal) and we write u=v.
Chapter 10, 10.2, P524
BCABAC
Chapter 10, 10.2, P525
DEFINITION OF VECTOR ADDITION If u and v are vectors positioned so the initial point of v is at the terminal point of u, then the sum u + v is the vector from the initial point of u to the terminal point of v.
Chapter 10, 10.2, P525
Chapter 10, 10.2, P525
Chapter 10, 10.2, P525
Chapter 10, 10.2, P525
Chapter 10, 10.2, P525
DEFINITION OF SCALAR MULTIPLICATION If c is a scalar and v is a vector, then the scalar multiple cv is the vector whose length is │c│ times the length of v and whose direction is the same as v if c>0 and is opposite to v if c<0. If c=0 or v=0, then cv=0.
Chapter 10, 10.2, P526
Chapter 10, 10.2, P526
Notice that two nonzero vectors are parallel if they are scalar multiples of one another. In particular, the vector –v=(-1)v has the same length as v but points in the opposite direction. We call it the negative of v.
Chapter 10, 10.2, P526
By the difference u - v of two vectors we mean
u - v= u + (-v)
Chapter 10, 10.2, P526
Chapter 10, 10.2, P527
Chapter 10, 10.2, P527
Chapter 10, 10.2, P527
1. Given the points A(x1,y1,z1) and B(x2,y2,z2), the vector a with representation AB is
a=<x2-x1,y2-y1,z2-z1>
Chapter 10, 10.2, P527
Chapter 10, 10.2, P527
Chapter 10, 10.2, P527
The length of the two-dimensional vector a=<a1,a2> is
The length of the three-dimensional vector a=<a1,a2,a3> is
22
21 aaa
23
22
21 aaaa
Chapter 10, 10.2, P527
if a=<a1,a2> and b=<b1,b2>, then the sum is a + b=<a1+b1, a2+b2>To add algebraic vectors we add their components. Similarly, to subtract vectors we subtract components. From the similar triangles in Figure 15 we see that the components of ca are ca1 and ca2. So to multiply a vector by a scalar we multiply each component by that scalar.
Chapter 10, 10.2, P528
If a=<a1,a2> and b=<b1,b2>, then
Similarly, for three-dimensional vectors,
2211 , bababa 2211 , bababa
21,cacaca
332211321321 ,,,,,, babababbbaaa
332211321321 ,,,,,, babababbbaaa
321321 ,, cacacaaaac
Chapter 10, 10.2, P528
We denote by V2 the set of all two-dimensional vectors and by V3 the set of all three-dimensional vectors. More generally, we will later need to consider the set Vn of all n-dimensional vectors. An n-dimensional vector is an ordered n-tuple:
na‧‧‧aaa ,, 21
Chapter 10, 10.2, P528
PROPERTIES OF VECTORS If a, b, and c are vectors in Vn and c and d are scalars, then
1. a + b=b + a 2. a + (b - c)=( a + b )+ c
3. a+0=a 4. a+(-a)=0
5. c(a + b)= ca + cb 6. (c + d) a= ca + da
7. (cd) a=c (da) 8. la=a
Chapter 10, 10.2, P529
Chapter 10, 10.2, P529
Chapter 10, 10.2, P529
Chapter 10, 10.2, P529
Three vectors in V3 play a special role. Let
i=<1,0,0> j=<0,1,0> k=<0,0,1>
These vectors i ,j , and k are called the standard basis vectors.
Chapter 10, 10.2, P529
If a=<a1,a2,a3> , then we can write
Thus any vector in V3 can be expressed in terms of i, j, and K.
kajaiaa 321
Chapter 10, 10.2, P529
In two dimensions, we can write
a=<a1,a2>=a1i+a2j
Chapter 10, 10.3, P533
1.DEFINITION If a=<a1,a2,a3> and b=<b1,b2,b3> , then the dot product of a and b is the number a‧b given by
332211 babababa
Chapter 10, 10.3, P533
2. PROPERTIES OF THE DOT PRODUCT If a, b, and c are vectors in V3 and c is a scalar, then
1. 2.
3. 4.
5.
2aaa abba
cabacba )( )()()( cbabacbca
00 a
Chapter 10, 10.3, P534
3. THEOREM If θ is the angle between the vectors a and b, then
cosbaba
Chapter 10, 10.3, P534
6. THEOREM If θ is the angle between the nonzero vectors a and b, then
ba
bacos
Chapter 10, 10.3, P535
7. Two vectors a and b are orthogonal if and only if a‧b = 0.
Chapter 10, 10.3, P535
If S is the foot of the perpendicular from R to the line containing PQ, then the vector with representation PS is called the vector projection of b onto a and is denoted by prjoa b. (You can think of it as a shadow of b). The scalar projection of b onto a (also called the component of b along a) is defined to be numerically the length of the vector projection, which is the number │b│ cosθ, where θ is the angle between a and b. (See Figure 4.) This is denoted by compa b.
Chapter 10, 10.3, P535
Chapter 10, 10.3, P536
Chapter 10, 10.3, P536
Scalar projection of b onto a: compa b=
Vector projection of b onto a: proja b=
a
ba
aa
ba
a
a
a
ba2
Chapter 10, 10.4, P539
1. DEFINITION If a=<a1,a2,a3> and b=<b1,b2,b3> , then the cross product of a and b is the vector
1231132332 , babababababa
Chapter 10, 10.4, P539
A determinant of order 2 is defined by
bcadcd
ab
Chapter 10, 10.4, P539
A determinant of order 3 can be defined in terms of second-order determinants as follows:
21
213
31
312
32
32
321
321
321
cc
bba
cc
bba
cc
bba
ccc
bbb
aaa
Chapter 10, 10.4, P540
321
321
bbb
aaa
kji
aba
Chapter 10, 10.4, P541
5. THEOREM The vector a ╳ b is orthogonal to both a and b.
Chapter 10, 10.4, P541
Chapter 10, 10.4, P541
6. THEOREM If θ is the angle between a and b (so 0≤θ≤ ), then
sinbaba
Chapter 10, 10.4, P542
7. COROLLARY Two nonzero vectors a and b are parallel if and only if
0ba
Chapter 10, 10.4, P542
The length of the cross product a ╳ b is equal to the area of the parallelogram determined by a and b.
Chapter 10, 10.4, P542
Chapter 10, 10.4, P543
kji ikj jik
kij ijk jki
Chapter 10, 10.4, P543
ijji
Chapter 10, 10.4, P543
8. THEOREM If a, b, and c are vectors and c is a scalar, then
1.
2.
3.
4.
5.
6.
abba )()()( cbabacbca
cabacba )(
cbcacba )(
cbacba )()(
cbabcacba )()()(
Chapter 10, 10.4, P544
Chapter 10, 10.4, P544
321
321
321
)(
ccc
bbb
aaa
cba
Chapter 10, 10.4, P544
11. The volume of the parallelepiped determined by the vectors a, b, and c is the magnitude of their scalar triple product:
)( cbaV
Chapter 10, 10.5, P547
Chapter 10, 10.5, P547
Chapter 10, 10.5, P547
In 3-D space, a line L is determined if we know a point Po (xo , yo , zo) on L and the direction al vector V=<a, b, c>.
Let (x, y, z) be a point on L. Then
(1)Pararnetric equations of L:
2. x=xo + at y=yo + Lt z=zo + ct
(2)Symrnetric equations of L:
3. c
zz
b
yy
a
xx 000
Chapter 10, 10.5, P548
▓Figure 3 shows the line L inExample 1 and its relation to the given point and to the vector that gives its direction.
Chapter 10, 10.5, P549
▓Figure 4 shows the line L inExample 2 and the point P where it intersects the xy-plane.
Chapter 10, 10.5, P550
4. The line segment from ro to r1 is given by the vector equation
0 ≤ t ≤110)1()( trrttr
Chapter 10, 10.5, P550
A plane in space is determined by a point Po (xo ,yo ,zo ) in the plane and a vector n that is orthogonal to the plane. This orthogonal vector n is called a normal vector.
Chapter 10, 10.5, P551
Let be P( x, y, z) be an arbitrary point in the plane, and let r0 and r be the position vectors of P0 and P. We have
which can be rewritten as
vector equation of the plane.
0)( orrn
0rnrn
Chapter 10, 10.5, P551
We write n=<a, b, c>,r=<x, y, z> , and ro=<xo, yo, zo>. Then the vector equation (5) becomes
Or
Equation 7 is the scalar equation of the plane through Po(x0,y0,zo)with normal vector n=<a, b, c>.
0,,,, 000 zzyyxxcba
0)()()( 000 zzcyybxxa
Chapter 10, 10.5, P551
By collecting terms in Equation 7, we can rewrite the equation of a plane as
0 dczbyax
Chapter 10, 10.5, P553
plane: ax + by + cz + d = 0
Chapter 10, 10.5, P553
Refers to fiy11. Thus the formula for D can be written as
222
111
cba
dczbyaxD
Chapter 10, 10.6, P556
A quadric surface is the graph of a second-degree equation in three variables x, y, and z. The most general such equation is
where A, B, C‧‧‧J are constants, but by translation and rotation it can be broughtinto one of the two standard forms
or
0222 JIzHyGxFxzEyzDxyCzByAx
0222 JCzByAx 022 IzByAx
Chapter 10, 10.6, P558
FIGURE 6Vertical traces are parabolas; horizontal traces are hyperbolas. All traces are labeled with the value of k.
Chapter 10, 10.6, P558
FIGURE 7Traces moved to their correct planes
Chapter 10, 10.6, P558
FIGURE 8The surface z=y2-x2 is a hyperbolic paraboloid.
Chapter 10, 10.6, P559
TABLE 1 Graphs of Quadric Surfaces
All traces are ellipses.If a=b=c , the ellipsoid isa sphere.
12
2
2
2
2
2
c
z
b
y
a
x
Chapter 10, 10.6, P559
TABLE 1 Graphs of Quadric Surfaces
Horizontal traces are ellipses.Vertical traces are parabolas.The variable raised to thefirst power indicates the axisof the paraboloid.
2
2
2
2
b
y
a
x
c
z
Chapter 10, 10.6, P559
TABLE 1 Graphs of Quadric Surfaces
Horizontal traces arehyperbolas.Vertical traces are parabolas.The case where c<0 isillustrated.
2
2
2
2
b
y
a
x
c
z
Chapter 10, 10.6, P559
TABLE 1 Graphs of Quadric Surfaces
Horizontal traces are ellipses. Vertical traces in the planes x=k and y=k are hyperbolas if k≠0 but are pairs of lines if k= 0.
2
2
2
2
2
2
b
y
a
x
c
z
TABLE 1 Graphs of Quadric Surfaces
Horizontal traces are ellipses. Vertical traces are hyperbolas. The axis of symmetry corresponds to the variable whose coefficient is negative.
12
2
2
2
2
2
c
z
b
y
a
x
Chapter 10, 10.6, P559
TABLE 1 Graphs of Quadric Surfaces
Horizontal traces in z=k are ellipses if k>c or k<-c.Vertical traces are hyperbolas. The two minus signs indicate two sheets.
12
2
2
2
2
2
c
z
b
y
a
x
Chapter 10, 10.7, P561
A vector-valued function, or vector function, is simply a function whosedomain is a set of real numbers and whose range is a set of vectors.
Chapter 10, 10.7, P561
For every number t in the domain of r there is a unique vector in V3 denoted by r(t). If f(t), g(t), and h(t) are the components of the vector r(t), then f, g, and h are real-valued functions called the component functions of r and we can write
We use the letter t to denote the independent variable because it represents time in most applications of vector functions.
kthjtgitfthtgtftr )()()()(),(),()(
Chapter 10, 10.7, P561
If , then
provided the limits of the component functions exist.
)(),(),()( thtgtftr
)(lim),(lim),(lim)(lim thtgtftratatatat
Chapter 10, 10.7, P562
FIGURE 1C is traced out by the tip of a moving position vector r(t).
Chapter 10, 10.7, P563
Chapter 10, 10.7, P563
Chapter 10, 10.7, P565
The derivative r’ of a vector function r is defined in much the same way as for real valued functions:
h
trhtrtr
dt
drh
)()(lim)('
0
Chapter 10, 10.7, P565
The vector r’(t) is called the tangent vector to the curve defined by r at the point P, provided that r”(t) exists and r”(t)≠0. The tangent line to C at P is defined to be the line through P parallel to the tangent vector r’(t). We will also have occasion to consider the unit tangent vector, which is
)('
)(')(
tr
trtT
Chapter 10, 10.7, P565
Chapter 10, 10.7, P565
Chapter 10, 10.7, P566
4, THEOREM If
where f ,g, and h are differentiable functions, then
kthjtgitfthtgtftr )()()()(),(),()(
kthjtgitfthtgtftr )(')(')(')('),('),(')('
Chapter 10, 10.7, P566
Chapter 10, 10.7, P567
Chapter 10, 10.7, P567
FIGURE 13The curve r(t)=<1+t3,t2>is not smooth.
Chapter 10, 10.7, P568
5, THEOREM Suppose u and v are differentiable vector functions, c is a scalar, and f is a real-valued function. Then1.
2.
3.
4.
5.
6.
)(')(')]()([ tvtutvtudt
d
)(')]([ tcutcudt
d
)(')()()(')]()([ tutftutftutfdt
d
)(')()()(')]()([ tvtutvtutvtudt
d
)(')()()(')]()([ tvtutvtutvtudt
d
))((')('))](([ tfutftfudt
d
Chapter 10, 10.7, P569
kdtthjdttgidttfdttrb
a
b
a
b
a
b
a
)()()()(
Chapter 10, 10.8, P572
The length of a plane curve with parametric equations x=f(t), y=g(t), a ≤ t ≤b, as the limit of lengths of inscribed polygons and, for the case where f’ and g’ are continuous, we arrived at the formula
b
a
b
adt
dt
dy
dt
dxdttgtfL 2222 )()(])('[)]('[
Chapter 10, 10.8, P572
Suppose that the curve has the vector equation r(t)=<f(t), g(t), h(t)>, a ≤ t ≤b, then it can be show that its length is
b
a
b
a
dtdt
dz
dt
dy
dt
dx
dtthtgtfL
222
222
)()()(
)]('[)]('[)]('[
Chapter 10, 10.8, P572
Notice that both of the arc length formulas (1) and (2) can be put into the more compact form
b
adttrL )('
Chapter 10, 10.8, P572
FIGURE 1The length of a space curve is the limit of lengths of inscribed polygons.
Chapter 10, 10.8, P573
▓Piecewise-smooth curves were introduced on page 567.
Chapter 10, 10.8, P573
Now we suppose that C is a piecewise-smooth curve given by a vector function r(t)= f(t)i + g(t)j+ h(t)k, a≤ t ≤ b, and C is traversed exactly once as increases from a to b. We define its arc length function s by
Thus s(t) is the length of the part of C between r(a) and r(t). (See Figure 3.)
t
a
t
adu
du
dz
du
dy
du
dxduurts 222 )()()()(')(
Chapter 10, 10.8, P574
FIGURE 4Unit tangent vectors at equally spaced points on C
Chapter 10, 10.8, P574
8. DEFINITION The curvature of a curve is
where T is the unit tangent vector.
Namdy,
ds
dTk
)('
)(')(
tr
trtT
Chapter 10, 10.8, P574
We use the Chain Rule (Theorem 10.7.5, Formula 6) to write
and
But ds/dt=│r’(t)│ from Equation 7, so
)('
)(')(
tr
tTtk
dt
ds
ds
dT
dt
dT
dtds
dtdT
ds
dTk
/
/
Chapter 10, 10.8, P575
10.THEOREM The curvature of the curve given by the vector function r is
3)('
)(")(')(
tr
trtrtk
Chapter 10, 10.8, P577
▓We can think of the normal vector as indicating the direction in which the curve is turning at each point.
Chapter 10, 10.8, P577
We can define the principal unit normal vector N(t) (or simply unit normal) as
The vector B(t)=T(t)╳N(t) is called the binormal vector. It is perpendicular to both T and N and is also a unit vector. (See Figure 6.)
)('
)(')(
tT
tTtN
Chapter 10, 10.8, P578
)('
)(')(
tr
trtT
)('
)(')(
tT
tTtN )()()( tNtTtB
3)('
)(")('
)('
)('
tr
trtr
tr
tT
ds
dTk
Chapter 10, 10.9, P580
)(')()(
lim)(0
trh
trhtrtv
h
Chapter 10, 10.9, P580
The speed of the particle at time t is the magnitude of he velocity vector, that is, │v(t)│.
rate of change of distance with respect to time
dt
dstrtv )(')(
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