MICHAEL SPEARING
APRIL 29, 2013
CALCULUS III FINAL EXAM MATERIAL
CHAPTER 12: VECTORS AND THE GEOMETRY OF SPACE
This chapter introduces vectors and coordinate systems for threedimensional space. This will be the
setting for our study of the calculus of two variables. Vectors provide particularly simple descriptions of
lines and planes in space.
12.1: THREE  DIMENSIONAL COORDINATE SYSTEM
THREEDIMENSIONAL COORDINATE SYSTEM




12.2: VECTORS
COMBINING VECTORS


COMPONENTS





12.3: THE DOT PRODUCT
THE DOT PRODUCT
PROPERTIES OF DOT PRODUCT:
a
⋅
a = a
2
a
⋅
b = b
⋅
a
a
⋅
( b + c ) = a
⋅
b + a
⋅
c
a
⋅
0 = 0
k
a
⋅
b =
k
( a
⋅
b ) = a
⋅
k
b
a
⋅
b = a b cos(
θ
)

Two vectors a and b are orthogonal if and only if a
⋅
b = 0

The dot product is a scalar, not a vector
PROJECTIONS
 Scalar projection of b onto a: comp
a
b = ( a
⋅
b )
 a 
 Vector projection of b onto a: proj
a
b = ( a
⋅
b )
( a
)
 a 
 a 
 Projection of Big onto Little
12.4: THE CROSS PRODUCT
THE CROSS PRODUCT
 The cross product of two vectors is a vector
Solved by doing the determinant of order 3
a
⨯
b =

i
j
k
 = < a
2
b
3
 a
3
b
2
, a
3
b
1
 a
1
b
3
, a
1
b
2
 a
2
b
1
>

a
1
a
2
a
3


b
1
b
2
b
3

 The vector a
⨯
b is orthogonal to both a and b
 a
⨯
b  = a b sin(
θ
)
 Two vectors are parallel if and only if a
⨯
b = 0
 The length of a
⨯
b is equal to the parallelogram formed by the two vectors
PROPERTIES OF THE CROSS PRODUCT
a
⨯
b =  b
⨯
a
k
a
⨯
b =
k
( a
⨯
b ) = a
⨯
k
b
a
⨯
( b + c ) = a
⨯
b + a
⨯
c
a
⋅
( b
⨯
c ) = ( a
⨯
b )
⋅
c
TRIPLE PRODUCTS
 Scalar triple product
a
⋅
( b
⨯
c )
 Volume of parallelepiped is magnitude of a dot b cross c
 The shape formed by the three vectors
V =  a
⋅
( b
⨯
c ) 
12.5: EQUATIONS OF LINES AND PLANES
LINES
 To find the equation of a line, we need a point on the line and the direction ( v )
r = r
o
+ tv
 Two vectors are equal if their parametric equations are equal
x = x
0
+ at
y = y
0
+ bt
z = z
0
+ ct
 Symmetric equations:
x  x
0
= y  y
0
= z  z
0
a
b
c
 Line segment from r
0
to r
1
:
r(t) = (1t)r
0
+ rt
 Parallel lines:
 If parametric equations are proportional, lines are parallel
 Intersecting lines:
 Define two parameters t and s
 Solve one dimension ( x y or z )of parametric equations for t( or s )
 Plug into another dimension to get values for t and s
 If z
t
= z
s
, lines intersect at the equations solved with values t and s
 Skew lines:
 lines that do not intersect but are also not parallel

PLANES
 To define a plane, we need a point on the plane and an orthogonal vector (normal vector)
 Normal vector:
n = (a,b,c)
 Equation of a plane:
a( x  x
0
) + b( y  y
0
) + c( z  z
0
) = 0
ax + by + cz + d = 0
Distance from a point to a plane:
 Point: P(x
1
, y
1
, z
1
)
 Plane: ax + by + cz + d = 0
 absolute value of top, not magnitude
D =  ax
1
+ by
1
+ cz
1
+ d 
√
( a
2
+ b
2
+ c
2
)
12.6: CYLINDERS AND QUADRATIC SURFACES
–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
CHAPTER 13: VECTOR FUNCTIONS
13.1: VECTOR FUNCTIONS AND SPACE CURVES
VECTOR FUNCTIONS AND SPACE CURVES
 A vector function is a function whose range is a set of vectors
r( t ) = < f(t), g(t), h(t) >
13.2 DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONS
DERIVATIVES
 If r( t ) = < f(t), g(t), h(t) >
 Then, r’( t ) = < f’(t), g’(t), h’(t) >
 Follows the same rules as regular di
ff
erentiation
DIFFERENTIATION RULES