Math 23 Final Exam Material - MICHAEL SPEARING APRIL 29 ...

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 three-dimensional 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
THREE-DIMENSIONAL 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) = (1-t)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