4 VECTORS. 4.0 Introduction. Objectives. Activity 1
|
|
- Jeffery Stevens
- 6 years ago
- Views:
Transcription
1 4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply y sclr; pprecite the geometricl representtion of vectors; e le to evlute nd interpret the sclr nd vector products. 4.0 Introduction Some physicl quntities, such s temperture or time, re completely specified y numer given in pproprite units e.g. 20 C, hours. Quntities of this sort, which hve only mgnitude, re referred to s sclrs; wheres quntities for which it is lso necessry to give direction s well s mgnitude re clled vectors. Exmples include wind velocity force displcement. In this text we will print vectors in old type; for exmple,, etc. Unfortuntely, when writing vectors you cnnot distinguish them from sclrs in this wy. The stndrd wy of writing vectors is to underline them; for exmple ~, ~, etc. It is very importnt for you to conform to this nottion. lwys rememer to underline your vectors, otherwise gret confusion will rise! ctivity 1 Mke list of sclr nd vector quntities, distinguishing etween them. 89
2 Chpter 4 Vectors 4.1 Vector representtion vector cn e represented y section of stright line, whose length is equl to the mgnitude of the vector, nd whose direction represents the direction of the vector. ll the lines shown opposite hve the sme mgnitude nd direction, nd so ll represent the sme vector. Sometimes the nottion is used to represent the vector. Modulus The modulus of vector is its mgnitude. It is written s, nd is equl to the length of the line representing the vector. Equl vectors The vectors nd re equl if, nd only if, = nd nd re in the sme direction. How re the vectors shown opposite, = nd = PQ, relted? Q P Since =, nd they re in opposite directions, we sy tht =. Multipliction of vector y sclr If λ is positive rel numer, then λ is vector in the sme direction s nd is of mgnitude λ. If λ = 0, then λ is 2 the zero vector 0, nd if λ < 0 then λ is the vector in the opposite direction to nd of mgnitude λ. Does the definition mke sense when λ = 1? 90
3 Chpter 4 Vectors ctivity 2 Drw ny vector on sheet of pper, nd then lso drw () () (c) (d) 2, 3, , 1 2 c ctivity 3 h d The digrm opposite shows numer of vectors. Express vectors c, d, e, f, g nd h in terms of or. 4.2 Vector ddition f e g If the sides nd C of tringle C represent the vectors p nd q, then the third side, C, is defined s the vector sum of p nd q; tht is = p, C = q C = p+ q p q p + q C This definition of vector ddition is referred to s the tringle lw of ddition. You cn then sutrct vectors, for simply mens + ( ). For exmple = C C. s n exmple, consider the displcement vectors. If you wlk in the direction North for 5 miles, then Est for 10 miles, you cn represent these two displcements s vectors, nd, s shown opposite. The vector ddition of this indeed is the finl displcement from. nd is, nd 5 10 Exmple In the tringle C, represents, C represents. If D is the midpoint of, express C, C nd DC in terms of nd. 91
4 Chpter 4 Vectors C = + C = + C C = C = DC = D + C = D + C ) = 1 2 = ( +) (since D is the midpoint of D C Exmple C CD is prllelogrm. represents nd C represents. If M is the midpoint of C, nd N is the midpoint of D, find M nd N in terms of nd, nd hence show tht M nd N re coincident. D C = + C = + so M = 1 ( 2 + ). Similrly, D = + D = + C = +, so N = 1 ( 2 + ) nd N = + N = + 1 ( 2 + ) = = 1 2 ( + ) = M. Since N = M, N nd M must e coincident. 92
5 Chpter 4 Vectors Exercise 4 1. Four points,,, nd C re such tht = 10, = 5, C = Find nd C in terms of nd nd hence show tht, nd C re colliner. 4. C is tetrhedron with =, =, C = c. C 2. CD is qudrilterl. Find single vector which is equivlent to () + C Hence deduce tht () C + CD + C + CD + D = If, nd c re represented y in the cue shown, find terms of, nd c, the vectors represented y other edges. (c) + C + CD, D G F nd F in H C E Find C, nd C hence show tht + C + C = 0. in terms of, nd c nd 5. In regulr hexgon CDEF, = nd C =. Find expressions for DE D in terms of nd., DC, D nd 6. Given tht + C = + D, show tht the qudrilterl CD is prllelogrm. D 4.3 Position vectors In generl, vector hs no specific loction in spce. However, if =, where is fixed origin, then is referred to s the position vector of, reltive to. Is the position vector unique? The use of position vectors for solving geometry prolems hs lredy een illustrted in the previous section. Here, you will see mny more exmples of this type, ut first n importnt result. Theorem If nd re points with position vectors nd nd point C divides nd in the rtio λ:µ, then C hs position vector µ + λ λ + µ ( ) 93
6 Chpter 4 Vectors Proof If C divides in the rtio λ :µ, then C = λ ( λ + µ )? C (nd C = µ ( λ + µ ) ) Hence since C C = nd λ ( λ + µ ) re in the sme direction. Thus C λ = λ + µ ( ) = λ ( λ + µ ) ( ) nd C = + C = + λ ( λ + µ ) ( ) = ( λ + µ λ )+λ λ +µ ( ) = µ+ λ, s required. ( λ + µ ) Note tht, in the specil cse λ = µ, when C is the midpoint of, then C = 1 ( 2 + ) C nd this result hs lredy een developed in the lst section. Exmple In the tringle, = nd =. If C divides the line in the rtio 1:2 nd D divides the line in the rtio 1:2, find DC nd hence show tht DC is prllel to. C D 94
7 Chpter 4 Vectors From the result in the theorem ove C = 2+ 3 = Clerly D = 1 3, so DC = D + C = = 2 3 Since DC is multiple of, it is in the sme direction s ; tht is, DC is prllel to. Exmple In the figure opposite, X nd Y re the midpoints of nd respectively. If = nd = find the position vector of the point Z, the intersection of X nd Y. X Z X = 1 nd Y = Thus X = X + Y = = 1 2 Since Z lies on X, for some sclr numer, sy t, XZ = t X = t 1 2 nd Z = X + XZ = t 1 2 = 1 2 ( 1 t)+t. 95
8 Chpter 4 Vectors Similrly, for some sclr numer s, using the fct tht Z lies on Y gives Z = s+ 1 2 ( 1 s). ut these vectors re equl, so equting coefficients of nd, 1 ( 2 1 t )= s 2s+t =1 t = 1 ( 2 1 s ) s+2t =1 s =t = 1 3. Hence Z = 1 3 ( + ) Exmple 4 1. In the digrm T = 2T = = C = 2 () Find in terms of nd (i) (ii) T (iii) C (iv) T (v) TC. () From (iv) nd (v), wht cn you deduce out, T nd C? 2. From n origin the points, nd C hve position vectors, nd 2 respectively. The points, nd re not colliner. The midpoint of is M, nd the point of trisection of C nerer to is T. Drw digrm to show,,, C, M, T. Find, in terms of nd, the position vectors of M nd T. Use your results to prove tht, M nd T re colliner, nd find the rtio in which M divides T. 3. The vertices, nd C of tringle hve position vectors,, c respectively reltive to n origin. The point P is on C such tht P:PC = 3:1; the point Q is on C such tht CQ: Q = 2:3; the point R is on produced such tht R: R = 2:1. T C 2 The position vectors of P, Q nd R re p, q nd r respectively. Show tht q cn e expressed in terms of p nd r nd hence, or otherwise, show tht P, Q nd R re colliner. Stte the rtio of the lengths of the line segments PQ nd QR. 4. The points, nd C hve position vectors,, c respectively referred to n origin. () Given tht the point X lies on produced so tht : X = 2:1, find x, the position vector of X, in terms of nd. () If Y lies on C, etween nd C so tht Y: YC = 1:3, find y, the position vector of Y, in terms of nd c. (c) Given tht Z is the midpoint of C, show tht X, Y nd Z re colliner. (d) Clculte XY: YZ. 5. The position vectors of three points, nd C reltive to n origin re p, 3q p, nd 9q 5p respectively. Show tht the points, nd C lie on the sme stright line, nd stte the rtio : C. Given tht CD is prllelogrm nd tht E is the point such tht D = 1 3 DE, find the position vectors of D nd E reltive to. 96
9 Chpter 4 Vectors 4.4 Components of vector In this section, the ide of unit vector is first introduced. This is vector which hs unit mgnitude. So if is ny vector, its mgnitude is, nd (provided tht 0) the vector â = is unit vector, prllel to. Introducing crtesin xes xyz in the usul wy, unit vectors in the direction x, y nd z re represented y z i, j nd k. Suppose P is ny point with coordintes (x, y, z) reltive to s illustrted. i k j y Wht is the mgnitude nd direction of the vector x = xi? Clerly = xi ( = x) x z D nd + = xi+ C = xi+ yj nd + + P = xi+ yj+ D = xi+ yj+ zk Wht is the vector + + P? x P (x, y, z) r k j i C y Thus r= P = xi+ yi+ zk This vector is often written s 3 1 column mtrix x r= y z nd the nottion r= ( x, y, z) is lso sometimes used. 97
10 Chpter 4 Vectors ctivity 4 y For vectors with two dimensions, using unit vector i nd j s shown opposite, express,, c, d nd e in terms of i nd j. j c From the ctivity ove, you should hve shown tht, for exmple, c= i+ jnd e= 2i+ j. Then the vector c+ e is simply 3i+ 2j: dding vectors in form is just mtter of dding components. For vectors in 2 dimensions, in generl r= P = xi+ yj f i d e x nd its mgnitude is given y the length P, where P = x 2 + y 2 r = x 2 + y 2 y y P (x, y) Exmple If = i+ 3j, = 2i 5j, c= 2i+4j, find j x () the component form of the vectors (i) + (ii) + c (iii) (iv) + c (v) 3+ 2; i x () the mgnitude of the vectors in (); (c) unit vectors in the direction of + nd + c. () (i) + = ( i+ 3j)+ ( 2i 5j)=3i 2j (ii) + c= ( 2i 5j)+ ( 2i+4j)=0i j= j (iii) = ( i+ 3j) ( 2i 5j)= i+8j (iv) + c= ( + ) c=3i 2j ( 2i+4j)=5i 6j (v) 3+ 2 = 3( i+ 3j)+22i 5j ( )=3i+9j+4i 10 j= 7i j () (i) + = ( 2) 2 = 9+4 = 13 (ii) + c = 1 (iii) = ( 1) = 65 (iv) + + c = 25 + ( 6) 2 = 61 98
11 Chpter 4 Vectors (v) 3+ 2 = ( 4) 2 = 65 (c) If n = +, ˆn = 1 ( 3i 2j 13 )= 3 13 i 2 13 j If n = + c, ˆn = 1 ( 1 j)= j For vectors in 3 dimensions, the position vector of the point P with coordintes (x, y, z) is given y r= xi+ yj+ zk nd its mgnitude is given y r = x 2 + y 2 + z 2 Why is this result true? You cn perform lgeric opertions in the usul wy; for exmple, if thus = 3i+ 2j+ k = i 2j+ k + = ( 3i+ 2 j+ k)+ ( i 2j+k) = 4i+ 2k = ( 3i+ 2 j+ k) ( i 2j+k) = 2i+ 4j = = 14 = ( 2 2 )+1 2 = 6 Note tht two vectors, nd re prllel if = λ for some non-zero λ ; furthermore, if λ > 0 they re ctully in the sme direction. lso note tht two vectors = 1 i+ 2 j+ 3 k nd = 1 i+ 2 j+ 3 k re equl if, nd only if, their components re equl, i.e. 1 = 1, 2 = 2, 3 = 3. 99
12 Chpter 4 Vectors ctivity 5 If = 2i+ j k, = i 2j+ 3k, which of the vectors elow re prllel to or? (i) (iii) (v) 2i j+ k (ii) 5i 10 j+15k 4i 2j 2k (iv) 6i+ 3j 3k 2i+ 4j 6k (vi) 4i 2j+ 2k Exercise 4C 1. Write in the form xi+ yj+ zk, the vectors represented y P if P is the point () (1, 1, 1) () (2, 1, 1) (c) (1, 1, 0) 2. P represents the vector r. Write down the coordintes of P if () r= 3i 4 j+ k () r= i+ 2 j k (c) r= 4k 3. Find the mgnitude of the vectors () = 6i+ 2 j+ 3k () = 2i j 2k 4. If = 2i+ 5j k, = i+ j+ 2k, c= 2i+3j k find () + (d) 5 +11c () ( + ) + c (c) + ( + c) 5. Show tht the points (4, 2, 16), (0, 10, 4 ) nd C( 6, 22, 14) re colliner. (c) c= + (d) d = lso find unit vectors in the direction of nd. 4.5 Products of vectors The 'lger' of vectors hs een developed in previous sections. You cn dd nd sutrct vectors, multiply vector y sclr ( λ), ut s yet not 'multiply' vectors. There re, in fct, two wys of multiplying vectors: one, the sclr product leding to sclr quntity; the other, the vector product, eing vector. Sclr product For ny two vectors nd, the sclr product, denoted y. is defined y. = cosθ Here is the modulus of, is the modulus of, nd θ is the ngle etween the direction of the two vectors. (Some texts refer to the sclr product s the 'dot' product, nd you sy ' dot ' for.) 100
13 Chpter 4 Vectors Exmple If = 2i, = 5j nd c= i+ j, find (). ().c (c).c (). = cos90 = = 0 (). c= c cos45 = ( since c = 2 ) 2 θ = 2 (c). c= c cos45 y = = 5 From the definition of the sclr product: (i) If nd re perpendiculr (s in () ove), then θ = 90 nd cosθ = 0, which gives. = 0. j c (ii) If, for non-zero vectors nd,. = 0, then cosθ = 0 cosθ = 0, since 0, 0 ; then θ = 90 nd nd re perpendiculr. i x To summrise, for non-zero vectors nd. = 0, perpendiculr lso it is cler tht. = 2 ctivity 6 Evlute the sclr products () () (c) (d) i.i, i. j, i.k j.i, j. j, j.k k.i, k. j, k.k ( i+ j).j, ( 2i+k).k 101
14 Chpter 4 Vectors Check, in (d), tht for exmple, ( 2i+ k).k=2i.k+k.k ssuming tht the sclr product lwys ehves in this nturl wy, deduce formul for. when nd re expressed in component form = 1 i+ 2 j+ 3 k, = 1 i+ 2 j+ 3 k You should hve found in ctivity 6 tht i.i= j. j= k.k = 1, (i, j, k re unit vectors) i. j= j.k = k.i= 0 (i, j, k re mutully perpendiculr) So if nd re expressed in component form So. = ( 1 i+ 2 j+ 3 k). ( 1 i+ 2 j+ 3 k) = 1 ( 1 i.i+ 2 i. j+ 3 i.k ) + 2 ( 1 j.i+ 2 j. j+ 3 j.k ) + 3 ( 1 k.i+ 2 k. j+ 3 k.k) = (using the results ove). = Exmple If = 2i+ j+ 3k, = 3i+j 2k, find. nd the cosine of the ngle etween nd.. = ( 2i+ j+ 3k). ( 3i+ j 2k) = 2 ( 3) ( 2)= 6+1 6= 11 So. = cosθ = cosθ = 11 cosθ =
15 Chpter 4 Vectors Exmple Show tht the vectors, = i+ 2 j k nd = 2i 2 j 2k, re perpendiculr.. = ( i+ 2j k).2i 2j 2k ( ) = ( 2)+ ( 1) ( 2) = = 0 Hence vectors nd re perpendiculr. ctivity 7 For the vector x = 3i+ 2j, y = i+ mj, determine the vlues of m for which () x is perpendiculr to y () x is prllel to y (c) the ngle etween x nd y is 30. Exmple If = 3i j+ 2k nd = mi 2j 3k, find the vlue of m for which nd re perpendiculr.. = ( 3i j+ 2k). ( mi 2j 3k) = 3m + ( 1) ( 2)+2( 3) = 3m = 3m 4 = 0 m = 4 3 So nd re perpendiculr when m =
16 Chpter 4 Vectors Vector product For ny two vectors, nd, the vector product, denoted y (or ) is defined y n turn from to = sinθ ˆn Here is the mgnitude of, is the mgnitude of, nd ˆn is unit vector, perpendiculr to oth nd nd in the sense of direction of liner motion when screw turns from to s illustrted. In the figure if nd re in horizontl plne, then n is verticl. direction of screw motion This implies tht, nd ˆn form right-hnded system similr to the i, j, k system. n Wht is the mgnitude of the vectors nd? Wht is the direction of the vector? To follow the direction of screw's motion turning from to gives the direction ˆn, tht is n = =. sin θ ( ˆn ) n turn from to So + = 0 nd the vector product is not, in generl, commuttive ( ). y Exmple If = 2i, = 5i nd c= i+ j, find () () (c) c (d) () = 2 5 sin90 k = 10k (k is perpendiculr to nd ) () c= 5 2 sin 45 ( k ) = 5k j c (c) c= 2 2 sin 45 ( k )=2k i x (d) = 2 2 sin 0 =0 =0 In similr wy, you cn see tht 104 i j= i j sin 90 k = k (since k is perpendiculr to i nd j, nd i, j, k form righthnded system).
17 Chpter 4 Vectors ctivity 8 Determine ll the vector products () i i, i j, i k () j i, j j, j k (c) k i, k j, k k You should hve found tht wheres nd i j= k, j k = i, k i= j j i= k, k j= i, i k = j i i= j j= k k = 0 gin ssuming tht ddition nd sutrction ehve in nturl wy, you cn use these results to find formul for in terms of their components. If then it cn e shown tht ctivity 9 = 1 i+ 2 j+ 3 k, = 1 i+ 2 j+ 3 k = ( )i+( )j+( )k Prove the formul ove for. Writing out n rry i j k work out n esy wy of rememering the formul for. Exmple If = i+ j+ k, = 2i+ 3j k, find. 105
18 Chpter 4 Vectors = ( i+ j+ k) ( 2i+3j k) = i ( 2i+ 3j+ k) j2i+3j k ( )+k ( 2i+3j k) = 20+ 3k ( j)+ ( 2k)+30 i+2j 3i 0 = 4i+3j+k lterntively you cn quickly evlute the vector product using the formul from ctivity 9; this gives = ( 1 3)i+ ( 2 ( 1) )j+( 3 2)k = 4i+3j+k Note tht if two vectors nd re prllel (or nti-prllel) then θ = 0 or π, nd = 0 ˆn = 0 Conversly, for non-zero vectors nd, Hence = 0 sin θ = 0 θ = 0, π = 0 = 0 or = 0 or, prllel Exmple If = i 3j+ 2k nd = 2i+6j 4k, find. Wht cn you sy out nd? = ( i 3j+ 2k) ( 2i+6j 4k) = i ( 2i+6j 4k) 3j ( 2i+6j 4k)+2k ( 2i+6j 4k) = 20+6k+4j 6k i 4j 12i 80 = 0 Hence nd re prllel. In fct you cn redily see tht =
19 Chpter 4 Vectors Exercise 4D 1. If = 2i+ j 2k nd = 3i+4k, find (). () the cute ngle etween these vectors (to the nerest degree) (c) unit vector which is perpendiculr to oth nd. 2. For nd in Question 1, find. Use this to find the ngle etween these vectors (to the nerest degree). 5. Given the vectors u = 3i+ 2 j nd v = 2i+ λj, determine the vlue of λ so tht () u nd v re t right ngles () u nd v re prllel (c) the cute ngle etween u nd v is The ngle etween the vectors i+ j nd i+ j+ λk is 45. Find the possile vlues of λ. 7. Given tht = 2i+ k, = i 2 j+ 3k clculte 3. Let = i 2 j+ k, = 2i+ j k. Given tht c= λ+ µ nd tht c is perpendiculr to, find the rtio of λ () the sclr product. () the vector product to µ. 8. The vectors u nd v re given 4. Find the vlue of λ for which the vectors 2i 3j+ k nd 3i+ 6 j+ λk re perpendiculr. y u = 2i j+ 2k, v = pi+ qk. Given tht u v = i+ sk, find p, q nd s. Find lso the cosine of the ngle etween u nd v. 4.6 pplictions In Chpter 5 you will see how vectors cn e used to solve prolems in 3-dimensionl spce concerned with lines nd plnes. Using vectors for these prolems is very convenient ut it is not the principl ppliction of vectors, which is for solving prolems in mechnics. These pplictions, nd tht of vector clculus to prolems in fluid mechnics, re eyond the scope of this text, ut if you pursue mthemtics in Higher Eduction you will pprecite their importnce. Here we look t some simpler pplictions. Projection of vector Let P e the foot of the perpendiculr from to the line. is clled the projection of onto the line. P Note tht P = cosθ, where = nd θ is ngle P P If i is unit vector in the direction, then.i= i cosθ = cosθ = P So Projection of onto the line =.i 107
20 Chpter 4 Vectors Exmple The force F = 10ĉ, where ĉ is unit vector in direction mking n ngle of 45 with ech of the positive x nd y xes. Find the projection of F on the x-xis. y F ĉ = 1 2 i+ j ( ) j 45 o x So F = 10 ( 2 i+ j) i nd F.i= 10 ( 2 i+ j).i = = 10 2 re of tringle For the tringle, let =, = So = sin θ θ P = ( sinθ) = P ( = se height ) So re of = 1 2 Exmple If is the point (5, 0), is the point (3, 0), find the re of the tringle. = 5i, = 3i+ 6j re of tringle = 1 2 = 1 2 5i ( 3i+ 6j) = k = k = =15 108
21 Chpter 4 Vectors Exmple The tringle C is defined y the points (0, 1, 2), (1, 5, 5) nd C(2, 3, 1). Find the re of C. = ( 1, 4, 3) nd C = ( 2, 2, 1). So you cn think of clculte C s s i+ 4j+ 3k nd C s 2i+ 2j k nd ( 4. ( 1) 3.2)i+ ( ( 1) )j+( )k or 10i+ 7j 6k. Hence re of C = 1 2 ( 10, 7, 6) = = Work done y force Work is done when force moves prticle through distnce. If F is the constnt force eing pplied, nd the prticle is moved from to where = d, then work done = F.d Exmple R lock slides down n inclined plne from to. Ignoring friction, the forces cting on the lock re its weight, W nd norml rection R. Clculte the work done y the forces in terms of h. h W W = Wj, d = hj+i W.d = Wj. ( hj+i)=wh nd R.d = 0 since R nd d re perpendiculr. So the work done is simply Wh. 109
22 Chpter 4 Vectors 4.7 Miscellneous Exercises 1. Find the sum of the vectors 2i+ j k, i+ 3j+ k, 3i+ 2 j. 2. Find the mgnitude of the vector = 3i 2 j+ 6k. 3. If ( + 2)i+ ( 1)j nd ( 1)i j re equl vectors, find the vlues of nd. 4. If λi 4 j is prllel to 2i 6 j, find the vlue of λ. 5. Find the unit vector in the direction of 2i j+ 2k. 6. Find the vector with mgnitude three nd prllel to 6i 3j+ 2k. 7. If C = 4i+14 j 5k, = i+ 2 j+ 7k, nd = 2i+ 6 j+ 37k, show tht the vectors C, C re prllel. Hence deduce tht the points, nd C re colliner. 8. QP = p, R = 3p, Q = q. M is the midpoint of QR. () Express P nd RQ in terms of p nd q. () Express MQ S in terms of p nd q. (c) If S lies on QP produced so tht QS = k QP, express MS in terms of p, q nd k. (d) Find the vlue of k if MS is prllel to Q. 9. Show tht 3i+ 7j+ 2k is perpendiculr to 5i j 4k. 10. The points, nd C hve coordintes (2, 1, 1), (1, 7, 3) nd ( 2, 5, 1) respectively. Find the re of the tringle C. 11. If L, M, N nd P re the midpoints of D, D, C nd C respectively, show tht LM is prllel to NP. P R M Q 12. The position vectors of points P nd R re 2i 3j+ 7k nd 4i+ 5j+ 3k respectively. Given tht R divides PQ in the rtio 2:1, find the position vector of Q if () R divides PQ internlly () R divides PQ externlly. 13. Given tht = i+ j, = 5i+ 7j, find the position vectors of the other two vertices of the squre of which nd re one pir of opposite vertices. 14. Given tht p = t 2 i+ ( 2t +1)j+ k nd ( )k where t is sclr q = ( t 1)i+ 3tj t 2 + 3t vrile, determine () the vlues of t for which p nd q re perpendiculr. () the ngle etween the vectors p nd q when t = 1, giving your nswer to the nerest 0.1. (E) 15. The point P hs position vector ( 1+ µ )i+ ( 3 2µ )j+ ( 4 + 2µ )k where µ is vrile prmeter. The point Q hs position vector 4i+ 2 j+ 3k. () The points P 0 nd P 1 re the positions of P when µ = 0 nd µ = 1 respectively. Clculte the size of ngle P 0 QP 1, giving your nswer to the nerest degree. () Show tht PQ 2 = ( 3µ 1) nd hence, or otherwise, find the position vector of P when it is closest to Q. (E) 16. Referred to fixed origin, the points, nd C hve position vectors i 2j+ 2k, 3i k nd i+ j+ 4k respectively. Clculte the cosine of the ngle C. Hence, or otherwise, find the re of the tringle C, giving your nswer to three significnt figures. (E) 110
On the diagram below the displacement is represented by the directed line segment OA.
Vectors Sclrs nd Vectors A vector is quntity tht hs mgnitude nd direction. One exmple of vector is velocity. The velocity of n oject is determined y the mgnitude(speed) nd direction of trvel. Other exmples
More informationSTRAND J: TRANSFORMATIONS, VECTORS and MATRICES
Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors
More informationAnalytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q.
1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples
More informationThings to Memorize: A Partial List. January 27, 2017
Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved
More informationLesson Notes: Week 40-Vectors
Lesson Notes: Week 40-Vectors Vectors nd Sclrs vector is quntity tht hs size (mgnitude) nd direction. Exmples of vectors re displcement nd velocity. sclr is quntity tht hs size but no direction. Exmples
More information9.4. The Vector Product. Introduction. Prerequisites. Learning Outcomes
The Vector Product 9.4 Introduction In this section we descrie how to find the vector product of two vectors. Like the sclr product its definition my seem strnge when first met ut the definition is chosen
More informationHigher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
More informationVectors , (0,0). 5. A vector is commonly denoted by putting an arrow above its symbol, as in the picture above. Here are some 3-dimensional vectors:
Vectors 1-23-2018 I ll look t vectors from n lgeric point of view nd geometric point of view. Algericlly, vector is n ordered list of (usully) rel numers. Here re some 2-dimensionl vectors: (2, 3), ( )
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More informationS56 (5.3) Vectors.notebook January 29, 2016
Dily Prctice 15.1.16 Q1. The roots of the eqution (x 1)(x + k) = 4 re equl. Find the vlues of k. Q2. Find the rte of chnge of 剹 x when x = 1 / 8 Tody we will e lerning out vectors. Q3. Find the eqution
More information2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the
More informationset is not closed under matrix [ multiplication, ] and does not form a group.
Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed
More informationCoordinate geometry and vectors
MST124 Essentil mthemtics 1 Unit 5 Coordinte geometry nd vectors Contents Contents Introduction 4 1 Distnce 5 1.1 The distnce etween two points in the plne 5 1.2 Midpoints nd perpendiculr isectors 7 2
More information1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE
ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check
More informationChapter 2. Vectors. 2.1 Vectors Scalars and Vectors
Chpter 2 Vectors 2.1 Vectors 2.1.1 Sclrs nd Vectors A vector is quntity hving both mgnitude nd direction. Emples of vector quntities re velocity, force nd position. One cn represent vector in n-dimensionl
More information1.2 What is a vector? (Section 2.2) Two properties (attributes) of a vector are and.
Homework 1. Chpters 2. Bsis independent vectors nd their properties Show work except for fill-in-lnks-prolems (print.pdf from www.motiongenesis.com Textooks Resources). 1.1 Solving prolems wht engineers
More informationVECTORS, TENSORS, AND MATRICES. 2 + Az. A vector A can be defined by its length A and the direction of a unit
GG33 Lecture 7 5/17/6 1 VECTORS, TENSORS, ND MTRICES I Min Topics C Vector length nd direction Vector Products Tensor nottion vs. mtrix nottion II Vector Products Vector length: x 2 + y 2 + z 2 vector
More informationMatrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24
Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition Mtrix lger Section.. Mtrix ddition, Sclr Multipliction nd rnsposition rectngulr rry of numers is clled mtrix ( the plurl is mtrices ) nd the
More informationThomas Whitham Sixth Form
Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos
More informationAlg. Sheet (1) Department : Math Form : 3 rd prep. Sheet
Ciro Governorte Nozh Directorte of Eduction Nozh Lnguge Schools Ismili Rod Deprtment : Mth Form : rd prep. Sheet Alg. Sheet () [] Find the vlues of nd in ech of the following if : ) (, ) ( -5, 9 ) ) (,
More information13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS
33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in
More information8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.
8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the
More informationPolynomials and Division Theory
Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the
More information1B40 Practical Skills
B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need
More informationDEFINITION The inner product of two functions f 1 and f 2 on an interval [a, b] is the number. ( f 1, f 2 ) b DEFINITION 11.1.
398 CHAPTER 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 11.1 ORTHOGONAL FUNCTIONS REVIEW MATERIAL The notions of generlized vectors nd vector spces cn e found in ny liner lger text. INTRODUCTION The concepts
More informationBridging the gap: GCSE AS Level
Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions
More informationCoalgebra, Lecture 15: Equations for Deterministic Automata
Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined
More informationCONIC SECTIONS. Chapter 11
CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round
More information7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus
7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e
More informationDEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS
3 DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS This chpter summrizes few properties of Cli ord Algebr nd describe its usefulness in e ecting vector rottions. 3.1 De nition of Associtive
More informationMATRIX DEFINITION A matrix is any doubly subscripted array of elements arranged in rows and columns.
4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review MTRIX DEFINITION mtrix is ny douly suscripted rry of elements rrnged in rows nd columns. m - Column Revised /0 n -Row m,,,,,, n n mn ij nd Order
More informationMathematics. Area under Curve.
Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding
More informationVectors. Introduction. Definition. The Two Components of a Vector. Vectors
Vectors Introduction This pper covers generl description of vectors first (s cn e found in mthemtics ooks) nd will stry into the more prcticl res of grphics nd nimtion. Anyone working in grphics sujects
More information10 Vector Integral Calculus
Vector Integrl lculus Vector integrl clculus extends integrls s known from clculus to integrls over curves ("line integrls"), surfces ("surfce integrls") nd solids ("volume integrls"). These integrls hve
More informationExploring parametric representation with the TI-84 Plus CE graphing calculator
Exploring prmetric representtion with the TI-84 Plus CE grphing clcultor Richrd Prr Executive Director Rice University School Mthemtics Project rprr@rice.edu Alice Fisher Director of Director of Technology
More informationTriangles The following examples explore aspects of triangles:
Tringles The following exmples explore spects of tringles: xmple 1: ltitude of right ngled tringle + xmple : tringle ltitude of the symmetricl ltitude of n isosceles x x - 4 +x xmple 3: ltitude of the
More informationLecture 3: Curves in Calculus. Table of contents
Mth 348 Fll 7 Lecture 3: Curves in Clculus Disclimer. As we hve textook, this lecture note is for guidnce nd supplement only. It should not e relied on when prepring for exms. In this lecture we set up
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
More information13.4 Work done by Constant Forces
13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push
More information7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement?
7.1 Integrl s Net Chnge Clculus 7.1 INTEGRAL AS NET CHANGE Distnce versus Displcement We hve lredy seen how the position of n oject cn e found y finding the integrl of the velocity function. The chnge
More informationBases for Vector Spaces
Bses for Vector Spces 2-26-25 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything
More informationA B= ( ) because from A to B is 3 right, 2 down.
8. Vectors nd vector nottion Questions re trgeted t the grdes indicted Remember: mgnitude mens size. The vector ( ) mens move left nd up. On Resource sheet 8. drw ccurtely nd lbel the following vectors.
More informationVECTOR ALGEBRA. Chapter Introduction Some Basic Concepts
44 Chpter 0 VECTOR ALGEBRA In most sciences one genertion ters down wht nother hs built nd wht one hs estblished nother undoes In Mthemtics lone ech genertion builds new story to the old structure HERMAN
More informationThe practical version
Roerto s Notes on Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7 The Fundmentl Theorem of Clculus: The prcticl version Wht you need to know lredy: The theoreticl version of the FTC. Wht
More informationLesson 8.1 Graphing Parametric Equations
Lesson 8.1 Grphing Prmetric Equtions 1. rete tle for ech pir of prmetric equtions with the given vlues of t.. x t 5. x t 3 c. x t 1 y t 1 y t 3 y t t t {, 1, 0, 1, } t {4,, 0,, 4} t {4, 0,, 4, 8}. Find
More informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
More informationShape and measurement
C H A P T E R 5 Shpe nd mesurement Wht is Pythgors theorem? How do we use Pythgors theorem? How do we find the perimeter of shpe? How do we find the re of shpe? How do we find the volume of shpe? How do
More informationI1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3
2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is
More information/ 3, then (A) 3(a 2 m 2 + b 2 ) = 4c 2 (B) 3(a 2 + b 2 m 2 ) = 4c 2 (C) a 2 m 2 + b 2 = 4c 2 (D) a 2 + b 2 m 2 = 4c 2
SET I. If the locus of the point of intersection of perpendiculr tngents to the ellipse x circle with centre t (0, 0), then the rdius of the circle would e + / ( ) is. There re exctl two points on the
More informationChapter 9 Definite Integrals
Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationDate Lesson Text TOPIC Homework. Solving for Obtuse Angles QUIZ ( ) More Trig Word Problems QUIZ ( )
UNIT 5 TRIGONOMETRI RTIOS Dte Lesson Text TOPI Homework pr. 4 5.1 (48) Trigonometry Review WS 5.1 # 3 5, 9 11, (1, 13)doso pr. 6 5. (49) Relted ngles omplete lesson shell & WS 5. pr. 30 5.3 (50) 5.3 5.4
More informationTorsion in Groups of Integral Triangles
Advnces in Pure Mthemtics, 01,, 116-10 http://dxdoiorg/1046/pm011015 Pulished Online Jnury 01 (http://wwwscirporg/journl/pm) Torsion in Groups of Integrl Tringles Will Murry Deprtment of Mthemtics nd Sttistics,
More informationChapter 14. Matrix Representations of Linear Transformations
Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn
More informationCoimisiún na Scrúduithe Stáit State Examinations Commission
M 30 Coimisiún n Scrúduithe Stáit Stte Exmintions Commission LEAVING CERTIFICATE EXAMINATION, 005 MATHEMATICS HIGHER LEVEL PAPER ( 300 mrks ) MONDAY, 3 JUNE MORNING, 9:30 to :00 Attempt FIVE questions
More information3. Vectors. Vectors: quantities which indicate both magnitude and direction. Examples: displacemement, velocity, acceleration
Rutgers University Deprtment of Physics & Astronomy 01:750:271 Honors Physics I Lecture 3 Pge 1 of 57 3. Vectors Vectors: quntities which indicte both mgnitude nd direction. Exmples: displcemement, velocity,
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationR(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of
Higher Mthemtics Ojective Test Prctice ook The digrm shows sketch of prt of the grph of f ( ). The digrm shows sketch of the cuic f ( ). R(, 8) f ( ) f ( ) P(, ) Q(, ) S(, ) Wht re the domin nd rnge of
More informationSeptember 13 Homework Solutions
College of Engineering nd Computer Science Mechnicl Engineering Deprtment Mechnicl Engineering 5A Seminr in Engineering Anlysis Fll Ticket: 5966 Instructor: Lrry Cretto Septemer Homework Solutions. Are
More informationMath 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions
Mth 1102: Clculus I (Mth/Sci mjors) MWF 3pm, Fulton Hll 230 Homework 2 solutions Plese write netly, nd show ll work. Cution: An nswer with no work is wrong! Do the following problems from Chpter III: 6,
More informationSimplifying Algebra. Simplifying Algebra. Curriculum Ready.
Simplifying Alger Curriculum Redy www.mthletics.com This ooklet is ll out turning complex prolems into something simple. You will e le to do something like this! ( 9- # + 4 ' ) ' ( 9- + 7-) ' ' Give this
More informationSection 4: Integration ECO4112F 2011
Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic
More informationA study of Pythagoras Theorem
CHAPTER 19 A study of Pythgors Theorem Reson is immortl, ll else mortl. Pythgors, Diogenes Lertius (Lives of Eminent Philosophers) Pythgors Theorem is proly the est-known mthemticl theorem. Even most nonmthemticins
More informationdx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω.
Chpter 8 Stility theory We discuss properties of solutions of first order two dimensionl system, nd stility theory for specil clss of liner systems. We denote the independent vrile y t in plce of x, nd
More information2 Calculate the size of each angle marked by a letter in these triangles.
Cmridge Essentils Mthemtics Support 8 GM1.1 GM1.1 1 Clculte the size of ech ngle mrked y letter. c 2 Clculte the size of ech ngle mrked y letter in these tringles. c d 3 Clculte the size of ech ngle mrked
More informationP 1 (x 1, y 1 ) is given by,.
MA00 Clculus nd Bsic Liner Alger I Chpter Coordinte Geometr nd Conic Sections Review In the rectngulr/crtesin coordintes sstem, we descrie the loction of points using coordintes. P (, ) P(, ) O The distnce
More informationSimilarity and Congruence
Similrity nd ongruence urriculum Redy MMG: 201, 220, 221, 243, 244 www.mthletics.com SIMILRITY N ONGRUN If two shpes re congruent, it mens thy re equl in every wy ll their corresponding sides nd ngles
More informationIMPOSSIBLE NAVIGATION
Sclrs versus Vectors IMPOSSIBLE NAVIGATION The need for mgnitude AND direction Sclr: A quntity tht hs mgnitude (numer with units) ut no direction. Vector: A quntity tht hs oth mgnitude (displcement) nd
More informationBest Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
More informationMEP Practice Book ES19
19 Vectors M rctice ook S19 19.1 Vectors nd Sclrs 1. Which of the following re vectors nd which re sclrs? Speed ccelertion Mss Velocity (e) Weight (f) Time 2. Use the points in the grid elow to find the
More informationIn-Class Problems 2 and 3: Projectile Motion Solutions. In-Class Problem 2: Throwing a Stone Down a Hill
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Deprtment of Physics Physics 8T Fll Term 4 In-Clss Problems nd 3: Projectile Motion Solutions We would like ech group to pply the problem solving strtegy with the
More informationMATH1131 Mathematics 1A Algebra
MATH1131 Mthemtics 1A Alger UNSW Sydney Semester 1, 017 Mike Mssierer Mike is pronounced like Mich mike@unsweduu Plese emil me if you hve ny questions or comments Office hours TBA (week ), or emil me to
More information8Similarity UNCORRECTED PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8.
8.1 Kick off with S 8. Similr ojects 8. Liner scle fctors 8Similrity 8. re nd volume scle fctors 8. Review U N O R R E TE D P G E PR O O FS 8.1 Kick off with S Plese refer to the Resources t in the Prelims
More informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More information378 Relations Solutions for Chapter 16. Section 16.1 Exercises. 3. Let A = {0,1,2,3,4,5}. Write out the relation R that expresses on A.
378 Reltions 16.7 Solutions for Chpter 16 Section 16.1 Exercises 1. Let A = {0,1,2,3,4,5}. Write out the reltion R tht expresses > on A. Then illustrte it with digrm. 2 1 R = { (5,4),(5,3),(5,2),(5,1),(5,0),(4,3),(4,2),(4,1),
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More informationCM10196 Topic 4: Functions and Relations
CM096 Topic 4: Functions nd Reltions Guy McCusker W. Functions nd reltions Perhps the most widely used notion in ll of mthemtics is tht of function. Informlly, function is n opertion which tkes n input
More informationCHAPTER 6 Introduction to Vectors
CHAPTER 6 Introduction to Vectors Review of Prerequisite Skills, p. 73 "3 ".. e. "3. "3 d. f.. Find BC using the Pthgoren theorem, AC AB BC. BC AC AB 6 64 BC 8 Net, use the rtio tn A opposite tn A BC djcent.
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics
SCHOOL OF ENGINEERING & BUIL ENVIRONMEN Mthemtics An Introduction to Mtrices Definition of Mtri Size of Mtri Rows nd Columns of Mtri Mtri Addition Sclr Multipliction of Mtri Mtri Multipliction 7 rnspose
More informationChapter 1: Logarithmic functions and indices
Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4
More informationCHAPTER 10 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS. dy dx
CHAPTER 0 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS 0.. PARAMETRIC FUNCTIONS A) Recll tht for prmetric equtions,. B) If the equtions x f(t), nd y g(t) define y s twice-differentile function of x, then t
More informationLesson-5 ELLIPSE 2 1 = 0
Lesson-5 ELLIPSE. An ellipse is the locus of point which moves in plne such tht its distnce from fied point (known s the focus) is e (< ), times its distnce from fied stright line (known s the directri).
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More informationChapter 8.2: The Integral
Chpter 8.: The Integrl You cn think of Clculus s doule-wide triler. In one width of it lives differentil clculus. In the other hlf lives wht is clled integrl clculus. We hve lredy eplored few rooms in
More information2.4 Linear Inequalities and Interval Notation
.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or
More informationSection 14.3 Arc Length and Curvature
Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in
More informationTHREE-DIMENSIONAL KINEMATICS OF RIGID BODIES
THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES 1. TRANSLATION Figure shows rigid body trnslting in three-dimensionl spce. Any two points in the body, such s A nd B, will move long prllel stright lines if
More informationLecture 2e Orthogonal Complement (pages )
Lecture 2e Orthogonl Complement (pges -) We hve now seen tht n orthonorml sis is nice wy to descrie suspce, ut knowing tht we wnt n orthonorml sis doesn t mke one fll into our lp. In theory, the process
More informationAndrew Ryba Math Intel Research Final Paper 6/7/09 (revision 6/17/09)
Andrew Ryb Mth ntel Reserch Finl Pper 6/7/09 (revision 6/17/09) Euler's formul tells us tht for every tringle, the squre of the distnce between its circumcenter nd incenter is R 2-2rR, where R is the circumrdius
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationThe Algebra (al-jabr) of Matrices
Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering he lgebr (l-jbr) of Mtrices lgebr s brnch of mthemtics is much broder thn elementry lgebr ll of us studied in our high school dys. In sense
More informationLINEAR ALGEBRA APPLIED
5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nth-order
More informationChapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY
Chpter 3 MTRIX In this chpter: Definition nd terms Specil Mtrices Mtrix Opertion: Trnspose, Equlity, Sum, Difference, Sclr Multipliction, Mtrix Multipliction, Determinnt, Inverse ppliction of Mtrix in
More information( β ) touches the x-axis if = 1
Generl Certificte of Eduction (dv. Level) Emintion, ugust Comined Mthemtics I - Prt B Model nswers. () Let f k k, where k is rel constnt. i. Epress f in the form( ) Find the turning point of f without
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT
SCHOOL OF ENGINEERING & BUIL ENVIRONMEN MARICES FOR ENGINEERING Dr Clum Mcdonld Contents Introduction Definitions Wht is mtri? Rows nd columns of mtri Order of mtri Element of mtri Equlity of mtrices Opertions
More information3. Vectors. Home Page. Title Page. Page 2 of 37. Go Back. Full Screen. Close. Quit
Rutgers University Deprtment of Physics & Astronomy 01:750:271 Honors Physics I Lecture 3 Pge 1 of 37 3. Vectors Gols: To define vector components nd dd vectors. To introduce nd mnipulte unit vectors.
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationEdexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks
Edexcel GCE Core Mthemtics (C) Required Knowledge Informtion Sheet C Formule Given in Mthemticl Formule nd Sttisticl Tles Booklet Cosine Rule o = + c c cosine (A) Binomil Series o ( + ) n = n + n 1 n 1
More information