MICHAEL SPEARINGAPRIL 29, 2013CALCULUS III FINAL EXAM MATERIALCHAPTER 12: VECTORS AND THE GEOMETRY OF SPACEThis chapter introduces vectors and coordinate systems for three-dimensional space. This will be thesetting for our study of the calculus of two variables. Vectors provide particularly simple descriptions oflines and planes in space.12.1: THREE - DIMENSIONAL COORDINATE SYSTEMTHREE-DIMENSIONAL COORDINATE SYSTEM----12.2: VECTORSCOMBINING VECTORS--COMPONENTS-

----12.3: THE DOT PRODUCTTHE DOT PRODUCTPROPERTIES OF DOT PRODUCT:a⋅a = |a|2a⋅b = b⋅aa⋅( b + c ) = a⋅b + a⋅ca⋅0 = 0ka⋅b =k( a⋅b ) = a⋅kba⋅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 vectorPROJECTIONS- Scalar projection of b onto a: compab = ( a⋅b )

| a |- Vector projection of b onto a: projab = ( a⋅b )( a)| a || a |- Projection of Big onto Little12.4: THE CROSS PRODUCTTHE CROSS PRODUCT- The cross product of two vectors is a vectorSolved by doing the determinant of order 3a⨯b =|ijk| = < a2b3- a3b2, a3b1- a1b3, a1b2- a2b1>|a1a2a3||b1b2b3|- 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 PRODUCTa⨯b = - b⨯aka⨯b =k( a⨯b ) = a⨯kba⨯( b + c ) = a⨯b + a⨯ca⋅( b⨯c ) = ( a⨯b )⋅cTRIPLE PRODUCTS- Scalar triple producta⋅( b⨯c )- Volume of parallelepiped is magnitude of a dot b cross c- The shape formed by the three vectorsV = | a⋅( b⨯c ) |12.5: EQUATIONS OF LINES AND PLANESLINES- To find the equation of a line, we need a point on the line and the direction ( v )r = ro+ tv- Two vectors are equal if their parametric equations are equalx = x0+ aty = y0+ btz = z0+ ct- Symmetric equations:x - x0= y - y0= z - z0abc- Line segment from r0to r1:r(t) = (1-t)r0+ 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 zt= zs, 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 - x0) + b( y - y0) + c( z - z0) = 0ax + by + cz + d = 0Distance from a point to a plane:- Point: P(x1, y1, z1)- Plane: ax + by + cz + d = 0- absolute value of top, not magnitudeD = | ax1+ by1+ cz1+ d |√( a2+ b2+ c2)12.6: CYLINDERS AND QUADRATIC SURFACES

–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––CHAPTER 13: VECTOR FUNCTIONS13.1: VECTOR FUNCTIONS AND SPACE CURVESVECTOR FUNCTIONS AND SPACE CURVES- A vector function is a function whose range is a set of vectorsr( t ) = < f(t), g(t), h(t) >13.2 DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONSDERIVATIVES- If r( t ) = < f(t), g(t), h(t) >- Then, r’( t ) = < f’(t), g’(t), h’(t) >- Follows the same rules as regular differentiationDIFFERENTIATION RULES

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