# Vectors In The Plane Homework Stu Schwartz Function

**Unformatted text preview: **1 28 Chapter 3 Euclidean Vector Spaces A/temative Notations for Up to now we have been writing vectors in R" using the notation Vectors (15) We call this the comma-delimited form. However, since a vector in R" is just a list of its n components in a speciﬁc order, any notation that displays those components in the correct order is a valid way of representing the vector, For example, the vector in (15) can be written as V=(vi,v2,...,vn) V 2 [vi U2 U71] which is called row-matrix form, or as vi 1)2 V = ( 17) v Uﬂj which is called column-matrix form. The choice of notation is often a matter of taste or convenience, but sometimes the nature of a problem will suggest a preferred notation. Notations (15), ( 16), and (17) will all be used at various places in this text. Concept Review :3 m a a e 7: e a Coordinates of a point Vector addition: parallelogram rule and triangle rule Geometric vector Direction 8 n—tuple Length Initial point Terminal point Equivalent vectors a Vector subtraction a n—space as Negathe 01' a VCCtOT 9 Vector operations in n—space: addition, subtraction, scalar multiplication Linear combination of vectors Scalar multiplication a Collinear (i.e., parallel) vectors Zero vector Components of a vector Skills 9 c a a 6 Sketch vectors whose initial and terminal points are given. - Find components of a vector Whose initial and terminal points are given. = Prove basic algebraic properties of vectors (Theorems 3.1.1 and 3.1.2). Perform geometric operations on vectors: addition, subtraction, and scalar multiplication. Perform algebraic operations on vectors: addition, subtraction, and scalar multiplication. Determine whether two vectors are equivalent. Determine whether two vectors are collinear. Exercise Set 3 In Exercises 1—2, draw a coordinate system (as in Figure 3, (a) v1 = (3, 6) (b) v2 (.4, _8) 110) and locate the points whose coordinates are given. (C) V3 : (_4! *3) (d) V4 I (31 4, 5) 1. (a) (3,4,5) (b) (—3,4,5) (c) (3,—4,5) (6) v5 = (3,3,0) (f) V6 : (_1’0’2) (d) (3,4, —5) (e) (—3, ~11, 5) (f) (~3,4, «5) 2. (a) (0,3, —3) (b) (3, —3,0) (c) (—3,0,0) 4. (a) v1 = (5,—4) (b) vz = (3,0) (d) (3,0, 3) (e) (Ova—3) (f) (0.3,0) (c) v3 = (0,—7) (d) v4 = (co—3) In Exercises 34, sketch the following vectors with the initial (e) v5 : (0., 4, — 1) (f) v6 = (2! 2, 2) points located at the origin. ln Exercises 5—6, sketch the following vectors with the initial points located at the origin. 5. 10. 11. 12. 13. 14. 15. i (a) . (a) . (a) . (a) P1(4, 8), P10, —5), P18, *7, 2), Pz(3, 7) Pg(“4, —7) P2(—2, 5, ~—4) (a) (b) (c) P1(—5,0), P2(—3,1) P,(0,0), P2(3,4) P1(—1,0,2), P2(0,—1,0) P,(2,2,2), P2(0,O,O) (b) (C) ( d) —~> 1n Exercises 7—8, ﬁnd the components of the vector P, P2. P1(3,5), P2(2,8) (b) P,(5,—2, 1), P3(2,4, 2) P1(—6.2). P,(0, 0, 0), P2(—4,—1) (b) P2(vl,6, 1) Find the terminal point of the vector that is equivalent to u = (1,2) and whose initial point is A(1, 1). , Find the initial point of the vector that is equivalent to u : (l, l, 3) and whose terminal point is B(—l, ~l, 2). (b) Find the initial point of the vector that is equivalent to u : (l, 2) and whose terminal point is 8(2, 0). (a) Find the terminal point of the vector that is equivalent to u = (l, 1, 3) and whose initial point is A(0, 2,0). 00 Find a nonzero vector u with terminal point Q(3, 0, ~5) such that (a) u has the same direction as V = (4, —~2, Al). (b) u is oppositely directed to V = (4, —2, —l). Find a nonzero vector u with initial point P(—1, 3, ~5) such that (a) u has the same direction as v = (6, 7, —3). (b) u is oppositely directed to V = (6, 7, ~3). Let u = (4, —1), V : (0,5), and w = (—3, *3). Find the components of (a)u+w (b)v—3u (c) 2(u ~ 5W) (d) 3V — 2(u + 2w) (e) ~3(w w 211 —l— v) (f) (~2u — V) « 5(v + 3w) Let u :(—3, 1,2), V : (4, 0, —8),. and w : (6, #1, ——4). Find the components of (b) 6n + 2V (d) 5(V — 411) (f) (2u — 7w) — (8V + u) (a) V — w (e) —V + u (e) ~3(V ~ 8w) Let u = (—3, 2, 1, O), V = (4, 7, —3, 2), and w = (5, —2, 8, 1). Find the components of 16. 17. 18. 19. 20, 21. 22. 23. 24. 25. 28. 3.1 Vectors in 2-Space, 3-Space, and n-Space 1 29 (a) V w W (b) 2n + 7V (c) —u + (v — 4w) ((1) 6(u ‘ 3V) (e) —V * w (f) (6V w W) — (4n + v) Let u, V, and w be the vectors in Exercise 15. Find the vector x that satisﬁes 5x — 2v 2 2(w — 5x). Letuz (5, —~1,0,3,—3),V: (—1,—1,7,2,0),and w 2 (#4, 2, —3, —5, 2). Find the components of (b) 2v + 3n (d) 5(—V + 4n — w) (a) w — u (c) —W + 3(V — u) (e) —2(3W+v)—l—(2u+w) (f) %(w—~5v+2u)+v Letu : (1,2, “3,5,0), v = (0,4, w~1, 1,2), and w = (7, 1, ~4, ~2, 3). Find the components of (a) V + w (b) 3(2u — V) (c) (3u # V) — (2n + 4w) Letu :(—3,1,2,4,4),v = (4, o, ~8, 1,2), and w = (6, ~l, —4, 3, ~33). Find the components of (a) V — W (b) 6n + 2v (c) (2u — 7w) g (8v + u) Let u, V, and w be the vectors in Exercise 18. Find the components ofthe vector x that satisﬁes the equation 3u+v—2w:3x+2w. Let u, v, and w be the vectors in Exercise 19. Find the components of the vector x that satisﬁes the equation 2u—v+x:7x+w. For what value(s) of t, if any, is the given vector parallel to u z (4, ~1)? (a) (81, —2) (0) (M2) Which of‘the following vectors in R6 are parallel to u z (—2, 1, 0,3, 5, 1)? (a) (4,2,0, 6, 10,2) (13) (4, —2, 0, —6, —lO, «2) (c) (0,0, 0,0, 0,0) Letu = (2, 1,0, 1,—l)andv : (—2, 3, l, O, 2).Findscalars a and b so that an —l— [7V : (—8, 8, 3, ~1, 7). (b) (81,21) Let u = (l, —1, 3,5) and V: (2, 1,0,—3). Find scalars a and!) so that an + bv : (1, ~4, 9, 18). Find all scalars c1, c1, and C3 such that ci(1, 2,0) +c'2(2, l, l)+c3(0,3, l) = (0,0,0) Find all scalars c1, C2, and C; such that 01(1. —1, O) +cz(3,2, l) +C3(0, 1,4) = (—1, l, 19) Find all scalars 6,, C2, and C3 such that Ci(—1. O, 2) +Cg(2, 2, —2) + C3(1,-—2, l):(—6,12,4) 1 30 Chapter 3 Euclidean Vector Spaces 29. Let u; = (~1,3,2,0), dz = (2, 0,4, ~1), [I3 = (7, 1,1,4), and U4 : (6, 3, l, 2). Find scalars cl, c2, C3, and c4 such that C1111 + 62112 +C3113 + mm = (0,5,6, ——3). Show that there do not exist scalars c1, C2, and C3 such that c1(l,0, 1, 0) +cz(l, O, —2, l) +C3(2,0, 1, 2) = (1, —2,2, 3) 30. Show that there do not exist scalars 6;, C2, and (:3 such that Cl(—2: 9, 6) + 62(—3,2, 1) + 63(1, 7, 5) = (0, 5, 4) 31. Consider Figure 3.1 . 12. Discuss a geometric interpretation of the vector 32. —> 1 —> —> 11 : OP1+ 5(0P2 *~ Let P be the point (2, 3, —2) and Q the point (7, —4, l). (a) Find the midpoint of the line segment connecting P and Q. (h) Find the point on the line segment connecting P and Q that is 37 of the way from P to Q. 33. Let P be the point (1,3, 7). If the point (4,0, —6) is the midpoint of the line segment connecting P and Q, what is Q? 34. 35. Prove parts (a), (c), and (d) of Theorem 31.1. 36. Prove parts (e)~(h) of Theorem 3.1.1. 37. Prove parts ((1)—(c) of Theorem 3.1.2. True-False Exercises ln parts (a)~(k) determine whether the statement is true or false, and justify your answer. (a) Two equivalent vectors must have the same initial point. (b) The vectors (a, b) and (a, b, 0) are equivalent. (0) If k is a scalar and v is a vector, then v and kv are parallel if and only if k 3 0. (d) The vectors v + (u + w) and (w + v) —l— u are the same. (e) lfu+v=u+w,thenv=w. (f) lfa and b are scalars such that an + bv = 0, then u and v are parallel vectors. (g) Collinear vectors with the same length are equal. (h) If (a, b, c) + (x, y, z) = (x, y, z), then (a, b, 6) must be the zero vector. (i) If k and m are scalars and u and v are vectors, then (k+m)(u+v) : ku+mv (j) If the vectors v and w are given, then the vector equation 3(2v—x) =5x—4w+v can be solved for x. (k) The linear combinations a1 v1 + azvz andbl v1 + bzvz can only be equal ifal 2 b1 and a; = b2. 3.2 ms; Damaganapalace is R5 ” in this section we will be concerned with the notions of length and distance as they relate to vectors. We will ﬁrst discuss these ideas in R2 and R3 and then extend them algebraically to R". Norm of a Vector In this text we will denote the length of a vector v by the symbol “VII, which is read as the norm of v, the length of v, or the magnitude of v (the term “norm” being a common mathematical synonym for length). As suggested in Figure 3.211a, it follows from the Theorem of Pythagoras that the norm of a vector (v1, 122) in R2 is M = 4v? + v; (1) Similarly, for a vector (v1, v2, D3) in R3, it follows from Figure 3.2. lb and two applica— tions of the Theorem of Pythagoras that W = (OR)2 + (RP)?- = mm2 + (QR)2 + (RP)2 = 2112+ u; + v32 and hence that M = M + v§ + vi (2) Motivated by the pattern of Formulas (1) and (2) we make the following deﬁnition. “Application of Dot Products to lSBN Numbers * Although the system has recently changed, most books published in the last 25 years have been assigned a unique 10-digit number called aniilntematioital Standard Book Number or ISBN. The ﬁrst nine digits'of this number are _split into three groups—the ﬁrst group rep— resenting the country or group of countries in which the book origi- nates, the second identifying the publisher, and the third assigned to thebdok title itself. The tenth and ﬁnal digit, called a check digit, is computed from the ﬁrst nine digits and is used to ensure that an electronic; transmission of the ISBN, say over the Internet, occurs without error. _ a .Toexplainhow this is done, regard the ﬁrstnine digits of the ISBN as a vector b in R9,- and let a be the vector ‘_ f ‘ a :V‘(l,'2,'3,4, 5, 6, 7, 8, 9) Then the check digitlc is computed using the following procedure: 1. _ Form the dot product a - b. “‘2. _ a - b by 11, thereby producing a remainder c that is an _ I integer between 0 and 10, inclusive. The check digit is taken to -: be c, with the proviso that c = 10 is written as X to avoid double digits. Concept Review “ Norm (or length or magnitude) of a vector e :2 e a Unit vector v Normalized vector 6 Standard unit vectors 9 Skills ' Compute the norm of a vector in R". 0 Determine whether a given vector in R” is a unit vector. ° Normalize a nonzero vector in R”. ° Determine the distance between two vectors in R”. Exercise Set 3.2 In Exercises l—Z, ﬁnd the norm of v, a unit vector that has the same direction as v, and a unit vector that is oppositely directed tov. l. (a) v: (4, —3) (c) v =(1,0,2,1,3) (b) v z (2, 2, 2) 2. (a) v:(—5, 12) (b) v: (1,—1,2) (c) V = (—2, 3, 3, —1) In Exercises 3—4, evaluate the given expression with u = (2, ~2, 3), v = (1, —3, 4), and w = (3, 6, —4). (b) Iiull+llvtl (d) i|3u — 5v + W“ 3. (a) iiu—l—vll (C) H—Zu+2vii Distance between points in R” Angle between two vectors in R” Dot product (or Euclidean inner product) of two vectors in R" CauchyASchwarz inequality 3.2 Norm, Dot Product, and Distance in R” 1 41 For example, the ISBN of the brief edition of Calculus, sixth edition, by Howard Anton is 0-471-15307-9 which has a check digit of 9. This is consistent with the ﬁrst nine digits of the ISBN, since ail):(1,2,3,4,5,6,7,8,9)-(0,4,7,1,1,5,3,0,7)::152 Dividing 152 by 11 produces a quotient of 13 and a remainder of 9, so the check digit is c = 9. If an electronic order is placed for a book with a certain ISBN, then the warehouse can use the above procedure to verify that the check digit is consistent with the ﬁrst . nine digits, thereby reducing the possibility of a costly shipping error. “ Triangle inequality 9 Parallelogram equation for Vectors ° Compute the dot product of two vectors in R". ‘ Compute the angle between two nonzero vectors in R". '1 Prove basic properties pertaining to norms and dot products (Theorems 32.1—32.3 and 3.2.5427). (b) in — vi to Hull — M 4. (a) Hu+v+wli (C) |i3Vil — 3|iVii In Exercises S~6, evaluate the given expression with u=(—2,—l,4,5),v=(3,1,~5,7), andw:(—6,2,1,1). 5- (a) li3u — 5v + wli (b) H3111! — 51lV1l+iiW’ii (C) ii-iiulivil 6- (a) liuil —2iiVii ~ 311W“ (c) H Ilu — vllwil (b) iiilli + lI—2Vil + Ii-3WII 7. Let v = (—2, 3, O, 6). Find all scalars k such that “ka : 5. 8. Let v : (1, 1, 2, «~3, 1). Find all scalars k such that likvil = 4. 1 42 Chapter 3 Euclidean Vector Spaces In Exercises 9461, ﬁnd 11 - V, 11 - u,_ and V . v. 9. (a u=(3,1,4), v:(2,2,—4) (b u=(l,l,4,6), v=(2,~2,3,—2) 10. (a) u: (1, 1,4,3), v:(—l,0,5, 1) (b) u=(2,_1,1.o,—2), v:(l,2,2,2,l) In Exercises H42, ﬁnd the Euclidean distance between a and v. 11. (a) u = (3,3,3), \’= (1,0,4) (1)) u = (0, —Z, ——1, 1), V = {—3, 2,4,4) (0) u = (3, ~3, —2,0, —3, 13,5), V: {—4, 1, e1, 5,0, —ll,4) 12. (a) u: (l,2,~3,0), v:(5, l,2,—2) (b) u :(2,—1,—4,1,0,6,—3.1). V: (-2, ~1,0,3,7,2, —5, 1) (c) u=(0,1,1, 1,2), v:(2, l,O,~l,3) 13. Find the cosine of the angle between the vectors in each part of Exercise 1 1, and then state whether the angle is acute, obtuse, or 90°. 14. Find the cosine of the angle between the vectors in each part of Exercise 12, and then state whether the angle is acute, obtuse, or 90°, 15. Suppose that a vector 21 in the .ry-plane has a length of9 units and points in a direction that is l20° counterclockwise from the positive x—axis, and a vector 1) in that plane has a length ofS units and points in the positive y—direction. Find a o b. 16. Suppose that a vector 21 in the xy—plane points in a direction that is 47° counterclockwise from the positive x-axis, and a vector b in that plane points in a direction that is 43° clock— wise from the positive x—axis. What can you say about the value ofa - b? In Exercises 17—18, determine whether the expression makes sense mathematically. If not, explain why. 17. (a) u ~ (V . w) (b) u - (v + w) (0) llu-Vll (d) (U‘V)— Hull 18. (a) 13111 - M (b) (u-V)—w (c)(u-v)~k (d)k-u 19. Find a unit vector that has the same direction as the given vector. (3) (~11, #3) (b) (1,7) 0» {—12.73 (d) (1,2,3,4.5) 20. Find a unit vector that is oppositely directed to the given vector. (21) {—12,—5) (b) (3,~3,—3) (c) <—6. 8) (d) 1—3. 1. 76, 3) 21. State a procedure for ﬁnding a vector of a speciﬁed length m that points in the same direction as a given vector v. 22. If llvil ': 2 and HWII : 3, what are the largest and smallest values possible for [iv ‘ wll? Give a geometric explanation of your results. 23. Find the cosine ofthe angle 6 between 11 and v. (a) u = (2, 3), v : (5, —7) (b) u : (~6, —2), v : (4, 0) (c) u = (l, —5,4), v : (3,3, 3) (d) u : (~2, 2, 3), v : (l, 7, «4) 24. Find the radian measure ofthe angle (9 (with 0 5 6 5 7;) be— tween u and v. (a) (l, ~7) and (21,3) (c) (—I, 1,0) and (O, —l, l) (b) (0,2) and (3, —3) (d) (l, ——l, 0) and (l, 0,0) In Exercises 25—26, verify that the Cauchy—Schwartz inequality holds. 25. (a) u=(3,2), v:(4,—1) (b) u:(—3, 1,0), v=(2,—1,3) (C) u=(0,2,2,1), v:(1,1,1,1) 26. (a) u:(4,l,l), v:(l,2,3) (b) u:(l,2, 1.2.3), v: (0, 1, 1,5,4) (c) u=(l,3,5,2,0,1), v=(0,2,4,1,3.5) 27. Let pO 2 (to, yo, Zo) andp = (x, y, z). Describe the set ofall points (x, y, z) for which llp w poll 2 l. 28. (a) Show that the components of the vector v : (1),, U3) in Figure Ex-28a are 1), = llvll c059 and v2 : HvH sin 9. (b) Let u and v be the vectors in Figure Ex-28b. Use the result in part (a) to ﬁnd the components of 411 ~ 5V. A .V (a) (b) Figure Ex-28 29. Prove parts (a) and (b) of Theorem 3.2.1. 30. Prove parts (a) and (c) of Theorem 3.2.3. 31. Prove parts (:1) and (e) ofTheorem 3.2.3. 32. Under what conditions will the triangle inequality (Theorem 3.2,5a) be an equality? Explain your answer geometrically. 33. What can you say about two nonzero vectors, u and V, that satisfy the equation llu + vi] 2 Hull + HVH? 34. (a) What relationship must hold for the point p = (a, b, c) to be equidistant from the origin and the xz-plane? Make sure that the relationship you state is valid for positive and negative values of a, b, and c. (b) What relationship must hold for the point p = (a, b, c) to be farther from the origin than from the xz—plane? Make sure that the relationship you state is valid for positive and negative values of o, b, and c True-False Exercises ln parts (a)—(j) determine whether the statement is true or false, and justify your answer. (a) If each component of a vector in R3 is doubled, the norm of that vector is doubled. (b) In R2, the vectors of norm 5 whose initial points are at the ori- gin have terminal points lying on a circle of radius 5 centered at the origin. 3.3 Orthogonality 3.3 Orthogonality 143 (c) Every vector in R” has a positive norm. (d) lfv is a nonzero vector in R”, there are exactly two unit vectors that are parallel to v. (e) If Hull 2 2, l|vl| : l, and u - V = 1, then the angle between it and V is 7r/3 radians. (f) The expressions (u - V) + W and u - (V + w) are both mean— ingful and equal to each other. (g) lfu-v:u-w,thenv:w, (h) lfu-V=O,theneitheru=00rv=0. (i) In R2, if u lies in the ﬁrst quadrant and v lies in the third quadrant, then u . V cannot be positive. (j) For all vectors u, V, and w in R”, we have llu + V+ Wll S llllll + NV” -|- “MI ln the last section we defined the notion of“angle” between vectors in R”. in this section we will focus on the notion of“perpendicularity.” Perpendicular vectors in f ” play an important role in a wide variety ofapplications. Orthogona/ Vectors Recall from Formula (20) in the previous section that the angle 6) between two nonzero vectors u and V in R” is defned by the formula <9=cos_1( 11W) llullllVll It follows from this that 6 = Jr/Z if and only if u o V = 0. Thus, we make the following deﬁnition. 3 i DEFINITION “3 Two nonzero vectors u and v in 1' " are said to be orthogonal (or r { perpendicular) ifu - V = 0. We will also agree that the zero vector in R” is orthogonal .,l to every vector in R”. A nonempty set of vectors in R” is called an orthogonal set if all pairs of distinct vectors in theset are orthogonal. An orthogonal set of unit vectors § i is called an orthonormal set. % EXANEEB t Orthogonal Vectors (a) Show that u z (—2, 3, 1,4) and V = (l, 2, 0, —l) are orthogonal vectors in R4. (b) Show that the set S = {L j, k} of standard unit vectors is an orthogonal set in R3, Solution (a) The vectors are orthogonal since u - V = (*2)(1)+(3)(2) +(1)(0)+(4)(—1) = 0 1 50 Chapter 3 Euclidean Vector Spaces The third distance problem posed above is to ﬁnd the distance between two parallel P a0 planes in R3. As suggested in Figure 3.3.7, the distance between a plane V and a plane V l W can be obtained by ﬁnding any point P0 in one of the planes, and computing the 1 distance between that point and the other plane. Here is an example. l 5 a Extﬁi‘ﬁt ma 8 Distance Between Parallel Planes W A F 3 3 7 Th d The planes igure . , e istance J _ ___ _ , 2 between the parallel planes V x + 2) ZZ 3 and 2x + 4y 44 and W is equal to the diStance are parallel since their normals, (l , 2, —2) and (2, 4, «4), are parallel vectors. Find the between P0 and W‘ distance between these planes. solution To ﬁnd the distance D between the planes, we can select an arbitrary point in one of the planes and compute its distance to the other plane. By setting y = z = O in the equation x + 2y — 2z 2 3, we obtain the point P0 (3, 0, 0) in this plane. From (16), the distance between P0 and the plane 2x + 4y — 4z 2 7 is D _ |2(3) + 4(0) + (“40(0) — 7] l /22 +42 + (_4)2 6 Concept Review « Orthogonal (perpendicular) -‘ Point—normal equations ‘ Vector component of u orthogonal to a 6 Theorem of Pythagoras VCC'COYS ° Vector form of a line @ Orthogonal set of vectors <2- Vcctor form of a plane <- Normal to a line 8v Orthogonal projection of u on a e Normal to a plane a Vector component of 11 along a ﬁkllls 5 Determine whether two vectors are orthogonal. a Compute the vector component of 11 along a and orthogonal to 3, Find the distance between a point and a line in R2 or R3. 6 Determine whether a given set of vectors forms an orthogonal set. a Find equations for lines (or planes) by using a normal 9 Find the distance between two parallel planes in R3, Vector and 3 POim 0“ the line (or Plane) Find the distance between a point and a plane. 0 Find the vector form of a line or plane through the origin, 0 Exercise Set 33 In Exercises 1—2, determine whether in and v are orthogonal vectors. 1. (a) u=(6,1,4), v=(2,o,—3) (b) u=(0,0,~l), v=(l, l, l) (c) u=(—6,0,4), v:(3, 1,6) (d) u : (2,4, —8), v = (5, 3, 7) in Exercises 3—4, determine whether the vectors form an or— thogonal set. 3. (a) v1 2 (2, 3), V2 2 (3,2) (b)V1=(—1,1),Vz = (1, l) (c) vl = (—2, l, 1), v2 = (1,0,2), v3 : (—2, —5, l) (d) v1 : (—3, 4, —l), v; = (1,2, 5), v3 = (4, ~3,0) 2. (a) u:(2,3), v: (5,4) 4. (a) v1 =(2,3), v2 = (—3,2) (b) u = (—6, «2), v: (4,0) (b) v, = (l, —2), v2 = (—2, l) (c) u:(1,—s,4), v=(3,3,3) (0) v1 =(l,0,l), v2: (1,1, 1), v3 =(—l,0, l) (d) u = (—2,2, 3), V = (l, 7, —4) (d) V1 : (2, —2, 1), V2 : (2, l, -—2), v3 = (l, 2, 2) 5. Find a unit vector that is orthogonal to both 11 = (1, 0, 1) and v z (0, 1, 1). 6. (a) Show that V = (a, b) and w = (—b, a) are orthogonal vectors. (1)) Use the result in part (a) to ﬁnd two vectors that are or- thogonal to v = (2, ~3). (c) Find two unit vectors that are orthogonal to (—3, 4). 7. DothepointsA(1, 1,1), B(—2, O, 3), and C(w3, —1, 1) form the vertices of a right triangle? Explain your answer. 8. Repeat Exercise 7 for the points A(3, 0,2), 8(4, 3, O), and C(8, l, ~1). In Exercises 9—12, ﬁnd a point-normal form of the equation of the plane passing through P and having n as a normal. 9. P(—1,3,—2);n:(~2,1,~1) 10. P(l,l,4); n:(l,9, 8) 11. P(2,0,0); n:(0,0,2) 12. P(0,0,0); n: (1,2,3) In Exercises 13—16, determine whether the given planes are parallel. 13. 4x~y+2z=5and7x~3y——4z=8 14. .rv4y—3z—220and3x—l2y49z—720 15. 2y=8x—4z+5andx = §z~},y 16. (—4, l, 2) - (x, y,z) : Oand (8, —2, —4) - (x,y,z) : 0 In Exerc'ses 17—18, determine whether the given planes are perpendicular. 17.3x—ye—z—420,x~2z=—1 18. x—2y~—3z=4, ~2x——5y+4z=ﬂl In Exercises 19—20, ﬁnd lprojaull. 19. (a) 11': (1, 42), a: (—4, 4) (b) u : (3, 0,4), a = (2, 3, 3) 20. (a) u : (5,6), 3 = (2, —1) (b) u=(3,—2, 6), a: (1,2, —7) In Exercises 21—28, ﬁnd the vector component of 11 along a and the vector component of u orthogonal to a. 21. 11: (6,2), a: (3,—9) 22. u: («1, —2), a: (~2,3) 23. u =(3,1,—7), a = (1,0, 5) 24. u=(1,0, 0), a: (4,3,8) 25. u =(l,1,l), a = (0,2, —1) 26. u = (2,0, l), a = (1,2,3) 27. u =(2,1, 1,2), 3 = (4, —4, 2, —2) 28, u: (5,0,—3,7), a: (2, l,—l,—l) 3.3 Orthogonality 151 In Exercises 29—32, ﬁnd the distance between the point and the line. 29. 4x+3y+4=0;(—3, 1) 30. .r-3y+2:0;(—l,4) 31. y:—4x+2; (2, —5) 32. 3x+y=5; (1,8) In Exercises 33—36, ﬁnd the distance between the point and the plane. 33. (3,1,-2); x+2y—2z=4 34. («—1,—1,2);2x+5y-6z=4 35. (~1,2,l);2x+3y—4Z=l 36. (0,3,—2); x—y—z=3 In Exercises 37—40, ﬁnd the distance between the given parallel planes. 37. 2x~y—z=5and—4x+2y+22 = 12 38. 3x«4y+'z=land6x~8y+2z=3 39. ~4x+y—3z=Oand8x—2y+6z=0 40. 2x~y+z=land2x~y+z=~l 41. Let i, j, and k be unit vectors along the positive x, y, and z axes ofa rectangular coordinate system in 3«space. va : (a, b, c) is a nonzero vector, then the angles a, ,8, and y between v and the vectors i, j, and k, respectively, are called the (lirectimz angles of v (Figure Ex—4l), and the numbers cos or, cos 1‘3, and cos y are called the direction cosines of v. (a) Show that cosot = a/llvll. (b) Find cos ,8 and cos y. (c) Show that v/llvll = (cos Cz’, cos )3, cos y). (d) Show that cos2 or + 0032 ,8 + cos2 y = l. «a Figure Ex~4t 42. Use the result in Exercise 41 to estimate, to the nearest de— gree, the angles that a diagonal of a box with dimensions 10 cm X 15 cm x 25 cm makes with the edges ofthe box. 43. Show that if V is orthogonal to both w, and w;, then v is orthogonal to klwl + kzwz for all scalars k, and kg. 44. Let u and v be nonzero vectors in 2— or 3—space, and let k = llull and] = llvll. Show that the vector w = In 4— kv bi- sects the angle between u and v. 1 52 Chapter 3 Euclidean Vector Spaces (d) I f a and b are orthogonal vectors, then for every nonzero vector _ j I h u, we have 46. Is it possd) e to ave projammjbw» : O 45. Prove part (a) of Theorem 3.3.4. Prolau : Molnar-7 (e) If a and u are nonzero vectors, then Explain your reasoning. projalpmlam» : pr0j3(u) __ I (f) If the relationship True-False txercrses In parts (a)—(g) determine whether the statement is true or false, and justify your answer. (a) The vectors (3, —l, 2) and (0, 0,0) are orthogonal. (g) For an vectors u and V, it is true that ' projau : projav holds for some nonzero vector a, then u 2 v. (b) If u and v are orthogonal vectors, then for all nonzero scalars “u + V” 2 “u” + “V” k and m, ku and mv are orthogonal vectors. (0) The orthogonal projection of u along a is perpendicular to the vector component of u orthogonal to a. 3.4 The Geometry of Linear Systems In this section we will use parametric and vector methods to study general systems of linear equations. This work will enable us to interpret solution sets of linear systems with n unknowns as geometric objects in R" just as we interpreted solution sets oflinear systems with two and three unknowns as points, lines, and planes in R2 and R3. Vector and Parametric In the last section we derived equations of lines and planes that are determined by a point Equations of Lines in R2 and a normal vector. However, there are other useful ways of specifying lines and planes. and R3 For example, a unique line in R2 or R3 is determined by a point x0 on the line and a nonzero vector v parallel to the line, and a unique plane in R3 is determined by a point x0 in the plane and two noncollinear vectors V1 and v; parallel to the plane. The best way to visualize this is to translate the vectors so their initial points are at X0 (Figure 3.4. l). Figure 3.4.2 Figure 3.4.1 Let us begin by deriving an equation for the line L that contains the point x0 and is parallel to v. Ifx is a general point on such a line, then, as illustrated in Figure 3.4.2, the vector x — x0 will be some scalar multiple of v, say X ~ x0 2 IV or equivalently x 2 X0 + W As the variable I (called a parameter) varies from —oc t0 36, the point X traces out the line L. Accordingly, we have the following result. ...

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Стекла очков блеснули, и его пальцы снова задвигались в воздухе. Он, как обычно, записал имена жертв. Контакты на кончиках пальцев замкнулись, и на линзах очков, подобно бестелесным духам, замелькали буквы.

ОБЪЕКТ: РОСИО ЕВА ГРАНАДА - ЛИКВИДИРОВАНА ОБЪЕКТ: ГАНС ХУБЕР - ЛИКВИДИРОВАН Тремя этажами ниже Дэвид Беккер заплатил по счету и со стаканом в руке направился через холл на открытую террасу гостиницы.

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