1. Vectors and Matrices
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1. Vectors and Matrices

Course: MATH 18.02, Spring 2008

School: MIT

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1. Vectors and Matrices 1A. Vectors Definition. A direction is just a unit vector. The direction of A is defined by A dir A = - , (A # 0); IAI it is the unit vector lying along A and pointed like A (not like -A). 1A-1 Find the magnitude and direction (see the definition above) of the vectors a) i + j + k b) 2 i - j + 2 k c) 3 i - 6 j - 2 k j 1A-2 For what value(s) of c will $i - + c k be a unit vector? +2j -...

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Vectors 1. and Matrices 1A. Vectors Definition. A direction is just a unit vector. The direction of A is defined by A dir A = - , (A # 0); IAI it is the unit vector lying along A and pointed like A (not like -A). 1A-1 Find the magnitude and direction (see the definition above) of the vectors a) i + j + k b) 2 i - j + 2 k c) 3 i - 6 j - 2 k j 1A-2 For what value(s) of c will $i - + c k be a unit vector? +2j - 2 k)/3. If its tail is a t 1A-3 a) If P = (1,3, -1) and Q = (0,1, I),find A = P Q , ( A ( ,and dir A. b) A vector A has magnitude 6 and direction ( i (-2,0, I), where is its head? 1A-4 a) Let P and Q be two points in space, and X the midpoint of the line segment P Q . Let 0 be an arbitrary fixed point; show that as vectors, O X = $(OP OQ) . + b) With the notation of part (a), assume that X divides the line segment P Q in the ratio r : s, where r s = 1. Derive an expression for O X in terms of O P and OQ. + 1A-5 What are the i j -components of a plane vector A of length 3, if it makes an angle of 30' with i and 60' with j . Is the second condition redundant? 1A-6 A small plane wishes to fly due north at 200 mph (as seen from the ground), in a wind blowing from the northeast at 50 mph. Tell with what vector velocity in the air it should travel (give the i j -components). 1A-7 Let A = a i b j be a plane vector; find in terms of a and b the vectors A' and A" resulting from rotating A by 90' a) clockwise b) counterclockwise. (Hint: make A ttie diagonal of a rectangle with sides on the x and y-axes, and rotate the whole rectangle.) c) Let i ' = (3 i 4j)/5. Show that i ' is a unit vector, and use the first part of the exercise to find a vector j ' such that i', j ' forms a right-handed coordinate system. 1A-8 The direction (see definition above) of a space vector is in engineering practice often given by its direction cosines. To describe these, let A = a i b j c k be a space vector, represented as an origin vector, and let a , p, and y be the three angles ( 5T)that A makes respectively with i , j ,and k . + + ++ a) Show that dir A = cos cr i called the direction cosines of A.) + cos Pj + cos y k . (The three coefficients are find the direction cosines b) Express the direction cosines of A in terms of a, b, c; ofthevector - i + 2 j + 2 k . c) Prove that three numbers t, u, v are the direction cosines of a vector in space if and only if they satisfy t2 + u2+ v 2 = 1. 2 E. 18.02 EXERCISES 1A-9 Prove using vector methods (without components) that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length. (Call the two sides A and B.) 1A-10 Prove using vector methods (without components) that the midpoints of the sides of a space quadrilateral form a parallelogram. 1A-11 Prove using vector methods (without components) that the diagonals of a parallelogram bisect each other. (One way: let X and Y be the midpoints of the two diagonals; show X = Y .) 1A-12* Label the four vertices of a parallelogram in counterclockwise order as OPQR. Prove that the line segment from 0 to the midpoint of PQ intersects the diagonal P R in a point X that is 113 of the way from P to R. (Let A = OP, and B = OR; express everything in terms of A and B.) 1A-13* a) Take a triangle PQR in the plane; prove that as vectors PQ +QR +RP = 0. b) Continuing part a), let A be a vector the same length as PQ, but perpendicular to it, and pointing outside the triangle. Using similar vectors B and C for the other two sides, prove that A B C = 0. (This only takes one sentence, and no computation.) ++ 1A-14* Generalize parts a) and b) of the previous exercise to a closed polygon in the plane which doesn't cross itself (i.e., one whose interior is a single region); label its vertices pi, P2,. .. ,Pn as you walk around it. 1A-15* Let Pi,. . . ,P, be the vertices of a regular n-gon in the plane, and 0 its center; show without computation or coordinates that OPl OP2+ ... OPn = 0, a) if n is even; b) if n is odd. + + 1B. Dot Product 1B-1 Find the angle between the vectors 1B-2 Tell for what values of c the vectors c i + 2 j - k and i -j +2k will a) be orthogonal b) form an acute angle 1B-3 Using vectors, find the angle between a longest diagonal PQ of a cube, and a) a diagonal PR of one of its faces; b) an edge PS of the cube. (Choose a size and position for the cube that makes calculation easiest.) 1B-4 Three points in space are P : (a, 1,- l ) , value(s) of a will PQR be Q : (0,1, I ) , R : (a, -1,3). For what a) a right angle b) an acute angle ? 1B-5 Find the component of the force F = 2 i - 2j +k in a) the direction +j - 4 b) the direction of the vector 3 i + 2j - 6 k . 1. VECTORS AND MATRICES 3 1B-6 Let 0 be the origin, c a given number, and u a given direction (i.e., a unit vector). Describe geometrically the locus of all points P in space that satisfy the vector equation In particular, tell for what value(s) of c the locus will be a) a plane b) a ray (i.e., a half-line) c) empty (Hint: divide through by (OPI.) ij 1B-7 a ) Verify that i ' = -and j ' = - perpendicular unit vectors that j are form a right-handed coordinate system + fi fi + b) Express the vector A = 2 i - 3j in the i ' j '-system by using the dot product. c) Do b) a different way, by solving for i and j in terms of i ' and j' and then substituting into the expression for A. 1B-8 The vectors i ' = a) Verify this. i+j+k .. fi perpendicular unit vectors that form a right-handed coordinate system. b) Express A = 2 i A ' j'x kt= i+j-2k are three mutually + 2j - k in this system (cf. 1B-7b) 1B-9 Let A and B be two plane vectors, neither one of which is a multiple of the other. Express B as the sum of two vectors, one a multiple of A , and the other perpendicular to A; give the answer in terms of A and B . (Hint: let u = dir A; what's the u-component of B?) 1B-10 Prove using vector methods (without components) that the diagonals of a parallelogram have equal lengths if and only if it is a rectangle. 1B-11 Prove using vector methods (without components) that the diagonals of a parallelogram are perpendicular if and only if it is a rhombus, i.e., its four sides have equal lengths. 1B-12 Prove using vector methods (without components) that an angle inscribed in a semicircle is a right angle. 1B-13 Prove the trigonometric formula: cos(O1 - 192) = cos O1 cos O2 + sine1 sin 82. (Hint: consider two unit vectors making angles 01 and 82 with the positive x-axis.) 1B-14 Prove the law of cosines: c2 = a 2 + b2 - 2abcosO by using the algebraic laws for the dot product and its geometric interpretation. 1B-15* The Cauchy-Schwarz inequality a) Prove from the geometric definition of the dot product the following inequality for vectors in the plane or in space: (*I 1A.BI I IAIIBI . Under what circumstances does equality hold? b) If the vectors are plane vectors, write out what this inequality says in terms of i j -components. 4 E. 18.02 EXERCISES c) Give a different argument for the inequality (*) as follows (this argument generalizes to n-dimensional space): i) for all values of t, we have ( A + t B ) . ( A +tB) 2 0 ; ii) use the algebraic laws of the dot product to write the expression in (i) as a quadratic polynomial in t ; iii) by (i) this polynomial has at most one zero; this implies by the quadratic formula that its coefficients must satisfy a certain inequality - what is it? 1C. Determinants 1C-1 Calculate the value of the determinants a) 1 : I:- b, I-; I;: 0 4 -1 2 using the Laplace expansion by the cofactors of: 12 1C-2 Calculate 3 -2 -1 a) the first row b) the first column 1C-3 Find the area of the plane triangle whose vertices lie at a) (O,O),(1,2), (1, -1); b) (1,2), (1, -11, (2,314 1C-4 Show that xl xf I x2 x3 x3 x1 I = (xi - x2)(x2 - x3)(x3 - x i ) . (This type of determinant is called a Vandermonde determinant.) 1C-5 a) Show that the value of a 2 x 2 determinant is unchanged if you add to the second row a scalar multiple of the first row. b) Same question, with "row" replaced by LLcolumn". 1C-6 Use a Laplace expansion and Exercise 5a to show the value of a 3 x 3 determinant is unchanged if you add to the second row a scalar multiple of the third row. 1C-'7 Let (xi, yl) and (x2,y2) be two unit vectors. Find the maximum value of the function 1C-8* The base of a parallelepiped is a parallelogram whose edges are the vectors b and c, while its third edge is the vector a. (All three vectors have their tail at the same vertex; one calls them "coterminal".) a) Show that the volume of the parallelepiped a b c is f a .( b x c) . b) Show that a . ( b x c) = the determinant whose rows are respectively the components of the vectors a,b,c. (These two parts prove (3), the volume interpretation of a 3 x 3 determinant. 1. VECTORS AND MATRICES 5 1C-9 Use the formula in Exercise 1C-8 to calculate the volume of a tetrahedron having as vertices (O,0, O), (0, -1,2), (0,1, -I), (1,2,1). (The volume of a tetrahedron is +(base) (height) .) 1C-10 Show by using Exercise 8 that if three origin vectors lie in the same plane, the determinant having the three vectors as its three rows has the value zero. ID. Cross Product I D - 1 Find A x B if a) A = i - 2 j + k , B=2i-j-k b) A = 2 i - 3 k , B=i+j-k. ID-2 Find the area of the triangle in space having its vertices a t the points P : (2,0, I ) , Q : (3, 1,O), R : ( - l , l , -1). 1D-3 Two vectors i ' and j ' of a right-handed coordinate system are to have the directions k . Find all three vectors i ' , j ', k ' . respectively of the vectors A = 2 i - j and B = i +2j + 1D-4 Verify that the cross product x does not in general satisfy the associative law, by showing that for the particular vectors i , i , j , we have ( i x i ) x j # i x ( i x j ) . 1D-5 What can you conclude about A and B a) if I A x B I = I A I I B I ; b) if I A x B I = A . B . 1D-6 Take three faces of a unit cube having a common vertex P; each face has a diagonal ending at P; what is the volume of the parallelepiped having these three diagonals as coterminous edges? 1D-7 Find the volume of the tetrahedron having vertices at the four points P : ( 1 , 0 , 1 ) , Q : ( - 1 , 1 , 2 ) , R:(O,O,2), S:(3,1,-1). Hint: volume of tetrahedron = ;(volume of parallelepiped with same 3 coterminous edges) 1D-8 Prove that A . ( B x C ) = (A x B ) . C , by using the determinantal formula for the scalar triple product, and the algebraic laws of determinants in Notes D. 1D-9 Show that the area of a triangle in the xy-plane having vertices a t (xi, yi), for 1 "1 Y1 1 i = 1,2,3, is given by the determinant - 2 2 y2 1 . DO this two ways: 2 3 Y3 1 a) by relating the area of the triangle to the volume of a certain parallelepiped b) by using the laws of determinants (p. L.l of the notes) to relate this determinant to the 2 x 2 determinant that would normally be used to calculate the area. 6 E. 18.02 EXERCISES 1E. Equations of Lines and Planes 1E-1 Find the equations of the following planes: a) through (2,0, -1) and perpendicular to i 2j - 2 k b) through the origin, (1,1, O), and (2, -1,3) c) through (1,0, I), (2, -1,2), (-1,3,2) d) through the points on the x, y and z-axes where x = a, y = b, z = c respectively (give the equation in the form Ax By + Cz = 1 and remember it) e) through (1,0,1) and (0,1,1) and parallel to i - j 2 k + + + 1E-2 Find the dihedral angle between the planes 2 s - y 1E-3 Find in parametric form the equations for + z = 3 and x + y + 22 = 1. + + a) the line through (1,0, -1) and parallel to 2 i - j 3 k b) the line through (2, -1, -1) and perpendicular to the plane x - y 22 = 3 c) all lines passing through (1,1,1) and lying in the plane x + 2y - z = 2 1E-4 Where does the line through (O,1, 2) and (2,0,3) intersect the plane x 1E-5 The line passing through (1,1, -1) intersects the plane 22 - y z = 1 at what point? + + 4y + z = 4? and perpendicular to the plane x + 2y - z = 3 + cz = d is given by 1E-6 Show that the distance D from the origin to the plane ax + by Idl the formula D = d a 2 b2 c2' ++ (Hint: Let n be the unit normal to the plane. and P be a point on the plane; consider the component of O P in the direction n.) 1E-7* Formulate a general method for finding the distance between two skew (i.e., nonintersecting) in lines space, and carry it out for two non-intersecting lines lying along the diagonals of two adjacent faces of the unit cube (place it in the first octant, with one vertex at the origin). (Hint: the shortest line segment joining the two skew lines will be perpendicular to both of them (if it weren't, it could be shortened).) IF. Matrix Algebra 1 Let A = a) B ( 2 -1 3 ,) , B= (-; ;) 1 -1 , C= (-; ;) 02 . Compute + C, B - C, 2B - 3C. b) AB, AC, BA, CA, BCT, CBT c) A(B + C ) , AB + AC; (B + C)A, BA + CA IF-2* Let A be an arbitrary m x n matrix, and let Ikbe the identity matrix of size k. Verify that I,A = A and AI, = A. IF-3 Find all 2 x 2 matrices A = ( i ) such that A2 (: :) . = 1. VECTORS AND MATRICES 7 1F-4* Show that matrix multiplication is not in general commutative by calculating for each pair below the matrix A B - BA: 1F-5 a) Let A = ( i).ComputeA2,A3,An. b)DothesameforA= (: i) . IF-6* Let A, A', B , B' be 2 x 2 matrices, and 0 the 2 x 2 zero matrix. Express in terms of these five matrices the product of the 4 x 4 matrices IF-7* Let A = A B - B A = 12. () = , B= (: !) . Show there are no values of a and b such that A = IF-8 a) If A 3 x 3 matrix A? b)* If A matrix A? (i) (3) (i) (i) (8) (!,) (8) (T), (1) (9) ( ) ( ) , , A = = , what is the A = , A = , what is the 1F-9 A square n x n matrix is called orthogonal if A . AT = I,. Show that this condition is equivalent to saying that a) each row of A is a row vector of length 1, b) two different rows are orthogonal vectors. 1 F - l o * Suppose A is a 2 x 2 orthogonal matrix, whose first entry is all = cos8. Fill in the rest of A. (There are four possibilities. Use Exercise 9.) 1F-11* Show that if A B and A B are defined, then a) ( A + B ) ~ = A ~ + B ~ , b) ( A B ) ~ = B ~ A ~ . + 1G. Solving Square Systems; Inverse Matrices For each of the following, solve the equation A x = b by finding A-' 1G-3 A = ( 1 -1 1 0 1 -1 -1 2 b= (9) . Solve A x = b b y f i n d i n g ~ - l 8 E. 18.02 EXERCISES 1G-4 Referring to Exercise 3 above, solve the system for the xi as functions of the yi. 1G-5 Show that (AB)-' = B-' A-', by using the definition of inverse matrix. 1G-6* Another calculation of the inverse matrix. If we know A-l, we can solve the system Ax = y for x by writing x = A-'y. But conversely, if we can solve by some other method (elimination, say) for x in terms of y, getting x = B y , then the matrix B = A-', and we will have found A-'. This is a good method if A is an upper or lower triangular matrix - one with only zeros respectively below or above the main diagonal. To illustrate: a) Let A = ( : -:) -1 1 -21 + + 3x3 = y1 22 ; find A-I by solving 2x2 - 2 3 = y2 23 for the xi = Y3 in terms of the yi (start from the bottom and proceed upwards). b) Calculate A-l by the method given in the notes. cos8 -sin8 corresponding to rotation of sin8 cos8 the x and y axes through the angle 8. Calculate Ag' by the adjoint matrix method, and explain why your answer looks the way it does. 1G-7* Consider the rotation matrix As = 1G-8* a) Show: A is an orthogonal matrix (cf. Exercise IF-9) if and only if A-' = AT b) Illustrate with the matrix of exercise 7 above. c) Use (a) to show that if A and B are n x n orthogonal matrices, so is AB. 1G-9* a) Let A be a 3 x 3 matrix such that IAl # 0. The notes construct a right-inverse A-', that is, a matrix such that A . A-' = I . Show that every such matrix A also has a left inverse B (i.e., a matrix such that BA = I.) (Hint: Consider the equation A ~ ( A ~ ) =' I; cf. Exercise IF-11.) b) Deduce that B = A-' by a one-line argument. (This shows that the right inverse A-' is automatically the left inverse also. So if you want to check that two matrices are inverses, you only have to do the multiplication on one side - the product in the other order will automatically be I also.) 1G-lo* Let A and B be two n x n matrices. Suppose that B = P-'AP for some invertible n x n matrix P . Show that Bn = P-'AnP. If B = In, what is A? 1G-11* Repeat Exercise 6a and 6b above, doing it this time for the general 2 x 2 matrix A= ( i), assuming I AJ # O. 1. VECTORS AND MATRICES 1H. Cramer's Rule; Theorems about Square Systems 1H-1 Use Cramer's rule to solve for x in the following: 3%-y+z= 1 -~+2y+z= 2 , 2-y+z=-3 X-y+z=O (b) x-z=l. -x+y+z=2 (a) 1H-2 Using Cramer's rule, give another proof that if A is an n x n matrix whose determinant is non-zero, then the equations Ax = 0 have only the trivial solution. 1H-3 a) For what c-value(s) will +23 = 0 2x1 + 2 2 + x3 = 0 -xl + cxz + 2x3 = 0 21 - x2 have a non-trivial solution? b) For what c-value(s) will ( -:)(;) =c (;) have a non-trivial solution? (Write it as a system of homogeneous equations.) c) For each value of c in part (a), find a non-trivial solution to the corresponding system. (Interpret the equations as asking for a vector orthogonal to three given vectors; find it by using the cross product.) d)* For each value of c in part (b), find a non-trivial solution to the corresponding system. 0-2y+z=o 1H-4* Find all solutions to the homogeneous system use the method suggested in Exercise 3c above. 1H-5 Suppose that for the system a1 # a l x + bly = cl = 0. Assume that we have a2x bny = c2 a2 0. Show that the system is consistent (i.e., has solutions) if and only if c2 = -el. a1 ; x+y-z=O 3~-3x+z=O + 1H-6* Suppose (A1= 0, and that x l is a particular solution of the system Ax = B. Show that any other solution x2 of this system can be written as x2 = x l xo, where xo is a solution of the system Ax = 0. + 1H-7 Suppose we want to find a pure oscillation (sine wave) of frequency 1passing through two given points. In other words, we want to choose constants a and b so that the function f(x) = a c o s x + bsinx , ) has prescribed values at two given x-values: f (XI)= y ~ f ( ~ 2 = y2. a) Show this is possible in one and only one way, if we assume that every integer n. b) If x2 = XI 22 # xl + nx, for + nx for some integer n, when can a and b be found? 10 E. 18.02 EXERCISES 1H-8* The method of partial fractions, if you do it by undetermined coefficients, leads to a system of linear equations. Consider the simplest case: ax b C x - 7-1 x - r l ) ( x - 7.2) + d +-x - T2 , (a, b given; c, d to be found); what are the linear equations which determine the constants c and d? Under what circumstances do they have a unique solution? (If you are ambitious, try doing this also for three roots ri, i = 1,2,3. Evaluate the determinant by using column operations to get zeros in the top row.) 11. Vector Functions and Parametric Equations 11-1 The point P moves with constant speed v in the direction of the constant vector a i b j . If at time t = 0 it is at (xo,yo), what is its position vector function r(t)? + 11-2 A point moves clockwise with constant angular velocity w on the circle of radius a centered a t the origin. What is its position vector function r(t), if a t time t = 0 it is at (a) (a, 0) (b) (O,a) 11-3 Describe the motions given by each of the following position vector functions, as t goes from -co to co. In each case, give the xy-equation of the curve along which P travels, and tell what part of the curve is actually traced out by P. 11-4 A roll of plastic tape of outer radius a is held in a fixed position while the tape is being unwound counterclockwise. The end P of the unwound tape is always held so the unwound portion is perpendicular to the roll. Taking the center of the roll to be the origin 0 , and the end P to be initially at (a, 0), write parametric equations for the motion of P . (Use vectors; express the position vector O P as a vector function of one variable.) 11-5 A string is wound clockwise around the circle of radius a centered at the origin 0;the initial position of the end P of the string is (a, 0). Unwind the string, always pulling it taut (so it stays tangent to the circle). Write parametric equations for the motion of P . (Use vectors; express the position vector O P as a vector function of one variable.) 11-6 A bow-and-arrow hunter walks toward the origin along the positive x-axis, with unit speed; a t time 0 he is at x = 10. His arrow (of unit length) is aimed always toward a rabbit hopping with constant velocity fiin the first quadrant along the line y = 22; at time 0 it is at the origin. a) Write down the vector function A(t) for the arrow at time t. b) The hunter shoots (and misses) when closest to the rabbit; when is that? 11-7 The cycloid is the curve traced out by a fixed point P on a circle of radius a which rolls along the x-axis in the positive direction, starting when P is at the origin 0 . Find the vector function O P ; use as variable the angle B through which the circle has rolled. (Hint: begin by expressing O P as the sum of three simpler vector functions.) 1. VECTORS AND MATRICES 11 1J. Differentiation of Vector Functions 15-1 1. For each of the following vector functions of time, calculate the velocity, speed Idsldtl, unit tangent vector (in the direction of velocity), and acceleration. a)eti+e-tj b)t2i+t3j C) ( 1 - 2 t 2 ) i + t 2 j + ( - 2 + 2 t 2 ) k 1 t 15-2 Let OP = - + i j be the position vector for a motion. 1+t2 1+t2 a) Calculate v , Jdsldtl, and T . b) At what point in the speed greatest? smallest? c) Find the xy-equation of the curve along which the point P is moving, and describe it geometrically. 15-3 Prove the rule for differentiating the scalar product of two plane vector functions: by calculating with components, letting r = X I i + yl j and s = x2 i + y2 j . 15-4 Suppose a point P moves on the surface of a sphere with center at the origin; let OP = r(t) = x(t) i y(t)j z(t) k . + + Show that the velocity vector v is always perpendicular to r two different ways: a) using the x, y, z-coordinates b) without coordinates (use the formula in 15-3, which is valid also in space). c) Prove the converse: if r and v are perpendicular, then the motion of P is on the surface of a sphere. 15-5 a) Suppose a point moves with constant speed. Show that its velocity vector and acceleration vector are perpendicular. (Use the formula in 15-3.) b) Show the converse: if the velocity and acceleration vectors are perpendicular, the point P moves with constant speed. 1 5-6 For the helical motion ~ ( t = a cos t i ) + a sin t j + bt k , a) calculate v, a, T , Idsldtl b) show that v and a are perpendicular; explain using 15-5 15-7 a) Suppose you have a differentiable vector function r(t). How can you tell if the parameter t is the arclength s (measured from some point in the direction of increasing t) without actually having to calculate s explicitly? +at) i +(yo +at) j ? b) How should a be chosen so that t is the arclength if r(t) = (XO c) How should a and b be chosen so that t is the arclength in the helical motion described in Exercise 15-6? 12 E. 18.02 EXERCISES 15-8 a) Prove the formula - u(t)r(t) = - r(t) dt d dt du + u(t)z. dr (You may assume the vectors are in the plane; calculate with the components.) b) Let r(t) = et cost i speed of this motion. + et sin t j , the exponential spiral. Use part (a) to find the 1J-9 A point P is moving in space, with position vector r = O P = 3costi + 5 s i n t j +4costk a) b) c) d) e) Show it moves on the surface of a sphere. Show its speed is constant. Show the acceleration is directed toward the origin. Show it moves in a plane through the origin. Describe the path of the point. 15-10 The positive curvature n of the vector function r(t) is defined by n = lgl. a) Show that the helix of 1J-6 has constant curvature. (It is not necessary to calculate s explicitly; calculate dT/dt instead and relate it to n by using the chain rule.) b) What is this curvature if the helix is reduced to a circle in the xy-plane? 1K. Kepler's Second Law 1K-1 Prove the rule (1) in Notes K for differentiating the dot product of two plane vectors: do the calculation using an i j -coordinate system. (Let r(t) = XI (t) i yl(t) j and s(t) = xz(t) i yz(t) j .) + + 1K-2 Let s(t) be a vector function. Prove by using components that -=0 ds dt s(t) = K , where K is a constant vector. 1K-3 In Notes K, by reversing the steps (5) - (8), prove the statement in the last paragraph. You will need the statement in exercise 1K-2.

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4. Line Integrals in the Plane4A. Plane Vector Fields4A-1 Describe geometrically how the vector fields determined by each of the following vector functions looks. Tell for each what the largest region in which F is continuously differentiable is.
MIT - MATH - 18.02
2. Partial Differentiation2A. Functions and Partial Derivatives2A-1 In the pictures below, not d l of the level curves are labeled. In (c) and (d), the picture is the same, but the labelings are different. In more detail: b) the origin is the level
MIT - MATH - 18.02
3. Double Integrals3A. Double Integrals in Rectangular Coordinates3A-1 Evaluate each of the following iterated integrals:(6x2 + 2y) dy dxb) l n I 2L n ( , sin t+t cos U)dt du3A-2 Express each double integral over the given region R as an it
MIT - MATH - 18.02
5. Triple Integrals5A. Triple integrals in rectangular and cylindrical coordinatesMiddle:iy+ iy2 yz] y=-1 +~ X Y dz Z dy ~ dx1=1+ z - (-1)=1XY+ 2zOuter: z+ z2]0 = 62b)Jd2 Jdfi JdxYInner: xy2z2] = x3y4 o1 Middle: ,x 4 y4
N.C. State - PHYSICS - 208m
c1. 2/2 points Last Response | Show Details All Responses Notes part score total submissions 1 1 1 2/2 2 1 1 1/2 - 2 2 -At location A there is an electric field in the direction shown by the orange arrow. This electric field is due to charged parti
N.C. State - PY - 208m
csponse | Show Details All Responses Notes part score total submissions 1 2 2 2/2 2 2 2 2/2 - 4 4 -Two vectors, and , are shown above. The diagrams below show various possible ways of combining and . A B CWhich diagram correctly shows graphical
North Texas - CHEM - 1410
SI: Thomas Stock Test 1 Review for Political Science 1040.001 1. How does Lasswell define politics? Who gets what, when, and how 2. What is the social contract? Social contract is an agreement between citizens and the government, in which the governe
Syracuse - IST - 616
Access and Access ControlIST616 Jian QinQuestions to ponder:Why do we need to select some data elements as access points while ignore the others and what are the criteria you consider as important for the selection? Each tag and indicator in MARC
Syracuse - IST - 616
System DesignIST616 Jian QinTwo types of users of library systemsA c q u is itio n R e c e iv in g P ro c e s s in g O rg a n iz a tio n In f o r m a t i o n S y s t e m (C a ta lo g s , in d e x e s , b ib lio g ra p h ie s , d ire c to rie s ,
Syracuse - IST - 616
Organizing Information in the Network EnvironmentMetadata Basics Jian QinOutline Metadata development overview Metadata records Metadata types and functions Metadata standards Metadata record creation and tools Metadata value space Referenc
Syracuse - IST - 616
Introduction to ClassificationIST616 Jian Qin09/19/08 IST616 - Classificaiton I 1Outline What is classification? Types of classification Classification structures How to classify library materials Library classification schemes09/19/08IS
Syracuse - IST - 616
Classification IIDDC & LCC number building Jian Qin IST616DDC number buildingFor more general introduction of DDC, please visit this animated and fun tour: http:/www.oclc.org/dewey/resources/tour/default.htm http:/www.oclc.org/dewey/resources/to
Syracuse - IST - 616
Encoding Formats for Bibliographic DataIST616 Jian QinWhats ahead What is MARC? Understanding MARC Relationship with AACR2 Fields, subfields, and indicatorsWhat is encoding syntax? Relationship with metadata standards HTML for Dublin
Syracuse - IST - 616
An Examination of FreeText Vs. Controlled Vocabulary DiscoursePrepared for IST 616 by Catherine Johnson and Curtis FerreeIntroduction and Background Debate in information science:free-text searching vs. controlled vocabulary searching The lite
Gustavus - E/M - 230
Problem 1-23A Importance of Cost Classification A.) Income Statements Sales Revenue (3000 x $14) Cost of Goods Sold Gross Margin Selling and Administrative Expense Net Income Balance Sheets Assets Cash Inventory Total Assets Stockholders' Equity Comm
Gustavus - E/M - 282
FACT SHEET -Economic Development Indicators-Profile Norms-Saudi ArabiaIndicator-Form 3Output and Agriculture: Agriculture value added %GDP Industry value added %GDP Services value added %GDP Food Prodn. Index, 1999-2001 = 100 Crop Prodn. Index, 199
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Illegal Immigration to the United StatesWhat should we do?Casey M. Dynan
Gustavus - E/M - 282
Cap3mex2ex2007Class Application Project #3Data for 2005 Indicator* Age dependency ratio (dependents to working-age population) Aid per capita (current US$) Consumer price index (2000 = 100) Exports of goods and services (% of GDP) External debt, t
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Leadership Within the Gustavus Greek CommunityTosin Cole Casey Dynan Adam Ingalsbe Alex Lair Camron LallierIs this what comes to mind?Agenda Intro Current situation at Gustavus SWOT analysis and data analysis Recommendations In depth recom
Cornell - CHEM - 1732
Welcome to Chemistry 2150!H. Floyd Davis Chapter 9: ThermochemistryLab Starts Next Week Read Chapter 9.1-9.2 Looking Ahead: Chapter 9.3-9.6System ClassificationOPENOPEN: exchanges both matter and energy with surroundingsSystem Classificati
Cornell - CHEM - 1732
Reactions in open containers (Const. P)For exothermic process q < 0: Define qp to be heat of reaction at constant pressure If a gas is produced, V > 0 Open Cup: Const. P Since w = -PV, work, w < 0E = qp + wEiEEfExperimental Determination o
S. Alabama - OPM - 101
Operations Management is:The management of systems or processes that create goods and/or provide service includes total quality management, worker involvement global competion and envirfomentla issures Value-added is the difference between the cost o
Georgia Tech - CEE - 4100
CEE-4100 Homework 2: Organization/Contracting/LawLast Name: First Name: Instructions: . .100 points total. Each question 2.5 points. Two bonus questions (each 1 point). Maximum 100 points. Due date Thursday, September 13, 3:00 P.M., Building SEB 3
CUNY Hunter - ENG - 320
Introduction to Media StudiesSections 94-103Seating AssignmentsSection # 94 95 96 97 98 99 100 101 102 103 Row B C D E F G H J K Llecture topic: key terms & concepts communication meaning perception culture and mediacommunicationthe wa
Penn State - HIST - 021
Ali Buschel: Paper # 1 - January 13, 2006 Penn State University Abington College HIS 021 Sec 002 American Civilization Since 1877 Dr. Prushankin Issue 2: Was John. D. Rockefeller A Robber Baron? Presented in Taking Sides, Madaras & SoRelle eds. Elev
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Ali Buschel: Paper #2 - 1/24/2006 Penn State University Abington College HIS 021 Sec 002 American Civilization Since 1877 Dr. Prushankin The Frontier in American History (1893) in For the Record: A Documentary History of the United States, Shi and M
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Ali Buschel: Paper #4 - 2/19/2006 Penn State University Abington College HIS 021 Sec 002 American Civilization Since 1877 Dr. PrushankinInterpreting Visual Sources: Photography and Progressive Reform in For the Record: A Documentary History of the
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History Of The United States From 18771. Waving the Bloody Shirt- Term used for the demagogy used in the United States in the 1860s and 70sby Reconstruction politicians to bring to mind the memories of the Civil War. The implication was that the De
Penn State - HIST - 021
Ali Buschel: Paper # 3 - February 13, 2006 Penn State University Abington College HIS 021 Sec 002 American Civilization Since 1877 Dr. Prushankin Issue 7: Did Yellow Journalism Cause the Spanish-American War? Presented in Taking Sides, Madaras & SoR
Virginia Tech - PHYS - 2205
Chapter 1: Introduction4. Scientific Notation and Significant FiguresSignificant figures : all the digits that are known accurately + one estimated digitRules Nonzero digits are always significant. Final or ending zeros to the right o
Virginia Tech - PHYS - 2205
Chapter 4: Motion with a Changing Velocity1. Motion along a line due to a constant net force If the net force is not zero, an object will accelerate. The acceleration of the object is constant, meaning constant in magnitude and in direction.
Virginia Tech - PHYS - 2205
Chapter 3: Acceleration and Newtons Second Law of Motion1. Position and displacementPosition : the location of an objectDisplacement : the direction and distance of the shortest path between an initial and final position:rrr r rf riE
Virginia Tech - PHYS - 2205
Physics 2205 - Practice Test 11. What is the angle between the vectors A and A when they are drawn from a common origin? A) 0 B) 90 C) 180 D) 270 E) 3602. A student adds two displacement vectors with magnitudes of 3.0 m and 4.0 m, respectively. Wh
Virginia Tech - PHYS - 2205
Chapter 2: Force1. ForceForce Loosely defined as pushing or pulling. An influence that may cause a body to accelerate Vector Quantity Unit : N = kg m/s2Long-range force and contact force See 2.6-2.8 Fundamental forces See 2.9Vec
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P1: OXT/SRB P2: xxx JWDD052-ANSWER JWDD052-Solomons-v2Printer: Hamilton April 24, 2007 19:5ANSWERS TO FIRST REVIEW PROBLEM SET1. (a)OH CH3 CH3 CH3 CH3 Hmethanide shift+OH2H A++CH3 CH3ACH3 CH3 CH3H2O +H2OCH32 Carbocation
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Chapter 13: Chemical Kinetics: Rates of Reactions16Chapter13:ChemicalKinetics:RatesofReactionsTeachingforConceptualUnderstandingKinetics is a topic with which students have many misconceptions. Among them are: (1) a catalyst gives the reactants
SUNY Stony Brook - CHE - 131
Chapter 16: Acids and Bases172As the reactants decompose, the concentrations of the products increase stoichiometrically, until they reach equilibrium concentrations. HA(aq) conc. init. (M) change conc. (M) eq. conc. (M) At equilibrium 0.040 x 0.
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Chapter 17: Additional Aqueous Equilibria709Chapter 17: Additional Aqueous Equilibria Questions for Review and ThoughtReview Questions1. The buffer capacity of a buffer solution is related to the concentrations of the buffers acid and base. The
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Chapter 13: Chemical Kinetics: Rates of Reactions54(b) Ea,reverse = Ea,forward H = (95 kJ mol1) (43 kJ mol1) = 138 kJ mol195 kJ/mol E 43.kJ/ ol m138 kJ/molreactantsproducts ReactionProgress(c) Ea,reverse = Ea,forward H = (55 kJ mol1) (
SUNY Stony Brook - CHE - 131
Chapter 16: Acids and Bases157Chapter16:AcidsandBasesTeachingforConceptualUnderstandingInstead of repeating the basic concepts of acids and bases that were introduced in Chapter 4, have students review that material on their own or assess their
SUNY Stony Brook - CHE - 131
Chapter 14: Chemical Equilibrium82Chapter14:ChemicalEquilibriumTeachingforConceptualUnderstandingThe most common misconception about equilibrium is that equal amounts of reactant and product are present when equilibrium is achieved. Many studen
SUNY Stony Brook - CHE - 131
Chapter 14: Chemical Equilibrium10452. Use a method similar to that described in the answer to Question 45(b). (conc. N2) = (conc. H2) = 1.00 mol N2O4/5.00 L = 0.200 M N2 (g) conc. initial (M) change conc. (M) equilibrium conc. (M) At equilibrium
SUNY Stony Brook - CHE - 131
Chapter 15: The Chemistry of Solutes and Solutions641Chapter 15: The Chemistry of Solutes and Solutions Questions for Review and ThoughtReview Questions1. Solubility in water, a highly polar covalent molecule, depends on the solute's ability to
SUNY Stony Brook - CHE - 131
CHE 132 S06 Practice Final Exam1. Question 1 deleted. 2. Which one of the following statements is not correct? a. The energy change of any system is equal to the heat gained by the system plus the work done on the system. b. Energy is a state funct
SUNY Stony Brook - CHE - 131
Practice Final Exam for Spring, 2006 CHE 132 General Chemistry II Stony Brook University1. When gasoline burns in the piston of an automobile engine, energy is transferred from the piston (the system) to the engine (the surroundings). Which one of
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P1: OXT/SRB JWDD052-15P2: QZI JWDD052-Solomons-v3Printer: Hamilton June 1, 2007 20:1915REACT+IONS OF AROMATIC COMPOUNDSSOLUTIONS TO PROBLEMSE H AHA+E H AHAE H+15.1AHAEEE15.2 The rate is dependent on the concentration
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P1: PCX/PBR JWDD052-AP2: PBU JWDD052-Solomons-v2Printer: Hamilton March 6, 2007 21:59ASPECIAL TOPIC Chain-Growth PolymersA.1 (a)Ph H Ph H H Ph H Ph Ph H H Ph Atactic polystyrene (Ph = C6H5) Ph H H Ph Ph H H Ph Ph H H Ph Syndiotactic polys
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P1: PCX/SRB JWDD052-17P2: xxx JWDD052-Solomons-v3Printer: Hamilton June 14, 2007 13:291717.1ALDEHYDES AND KETONES II. ENOLS AND ENOLATESSOLUTIONS TO PROBLEMSO OHCyclohexa-2,4-dien-1-one (keto form)Phenol (enol form)The enol form is
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P1: PBU/OVY JWDD052-HP2: PBU/OVYQC: PBU/OVYT1: PBUPrinter: Hamilton April 23, 2007 13:32JWDD052-Solomons-v3HH.1SPECIAL TOPIC Transition Metal Organometallic CompoundsSOLUTIONS TO PROBLEMSFe CO CO Cyclobutadiene iron tricarbonyl OC
SUNY Stony Brook - CHE - 321
P1: PBU/OVY JWDD052-GP2: PBU/OVYQC: PBU/OVYT1: PBUPrinter: Hamilton June 1, 2007 19:28JWDD052-Solomons-v3GSPECIAL TOPIC Electrocyclic and Cycloaddition ReactionsSOLUTIONS TO PROBLEMSG.1 According to the Woodward-Hoffmann rule for ele
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P1: PBU/OVY JWDD052-EP2: PBU/OVYQC: PBU/OVYT1: PBUPrinter: Hamilton April 30, 2007 11:16JWDD052-Solomons-v3ESPECIAL TOPIC AlkaloidsSOLUTIONS TO PROBLEMSE.1 (a) The rst step is similar to a crossed Claisen condensation (see Section 19
SUNY Stony Brook - CHE - 321
P1: PBU/OVY JWDD052-FP2: PBU/OVYQC: PBU/OVYT1: PBUPrinter: Hamilton April 23, 2007 13:43JWDD052-Solomons-v3FSPECIAL TOPIC ARYL HALIDES: THEIR USESSOLUTIONS TO PROBLEMSOF.1 Cl3C+OH HCl3COH+H + H2SO4Cl3CH+ HSO4OHCl3C
SUNY Stony Brook - CHE - 321
P1: QZI/PBR JWDD052-BP2: PBU JWDD052-Solomons-v3Printer: Hamilton April 30, 2007 11:15BSPECIAL TOPIC Step-Growth PolymersSOLUTIONS TO PROBLEMSOHB.1 (a)O2 cat.O +HNO3 cat.O HO OH O + N2OO (b) HO O Oheat 2 H2OO OH 2 NH3+NH4 O
SUNY Stony Brook - CHE - 321
P1: PCX JWDD052-24JWDD052-Solomons-v3Printer: Hamilton May 4, 2007 14:202424.1 (a)AMINO ACIDS AND PROTEINSSOLUTIONS TO PROBLEMSO HO NH3+ O (c) HO NH3+ OOO OH (b)O OO O NH2O O predominates at the isoelectric point rather thanO
SUNY Stony Brook - CHE - 321
P1: PCX JWDD052-25JWDD052-Solomons-v3Printer: Hamilton May 31, 2007 14:5925N N HNUCLEIC ACIDS AND PROTEIN SYNTHESISSOLUTIONS TO PROBLEMS25.1 Adenine:NH2 N N N N HNH H N N N N HNH N N H and so onGuanine: O N N H Cytosine: NH2 N N H
SUNY Stony Brook - CHE - 321
P1: OXT/SRB P2: PBY JWDD052-ANSWER JWDD052-Solomons-v3Printer: Hamilton April 30, 2007 11:14ANSWERS TO SECOND REVIEW PROBLEM SETThese problems review concepts from Chapters 1321. 1.Increasing acidity O (a) < OH CH < < MeO OH < O O OMe < O OH