Math 23 Final Exam Material - MICHAEL SPEARING CALCULUS III FINAL EXAM MATERIAL CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE This chapter introduces

# Math 23 Final Exam Material - MICHAEL SPEARING CALCULUS III...

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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
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