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1 7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6 = (" 6! 5", " 4! " 6," 5! 4 " = (!,6,! Note tht (!,6,! is perpendiculr to oth (,, nd (4,5,6. At first sight! seems n unwieldy formul ut use of the following mnemonic will possily help. Ech component of the cross product is otined y considering the following squre lock, multiplying digonlly dding the products nd sutrcting the products. i.e. the first component is i.e.!

2 7 Theorem. Proof! is perpendiculr to Let = (,, nd = (,, ( " = (!,!,! (,, = = 0 Similrly!By Theorem.(! is perpendiculr to.! is perpendiculr to. Theorem.! = sin! where! is the ngle etween nd. Proof We will show ( sin! =!. (note tht there will e no difficulty here with negtive signs since ech of,,! nd sin! is positive. (!! sin! =!! sin! =!! ( " cos! =!! # " (!! & %!! ( \$ % ' ( (Theorem.

3 74 ( ( ( ( ( ( ( ( ( ( (! = " " " = " " " = " = " = " = i.e.! = sin" In itself, Theorem. is not over-importnt ut it does give the very vlule corollry tht! is the re of the prllelogrm formed with nd s two djcent edges. Exmple Find re of ABC! where A is (-,,4, B is (,,5 nd C is (-,5, B (,,5 A (-,,4 C (-,5, D Are! =

4 75 AB = (5,,, AC = (,4,! AB " AC = (! 6,,9 But Are of # ABC = re of prllelogrm ABDC. = AB" AC = (! 6 ( (9 = 58 =.8 (pprox. Exmple Find vector perpendiculr to oth (,, nd (,,. Clerly (,,! (,, will suffice. (y Theorem. Exmple Find vector perpendiculr to the plne pssing through the points A (,-,, B(,0,- nd C (0,, B C A AB = (,,! AC = (!,,! Now note, since AB! AC will e perpendiculr to oth AB nd AC, then it will e vector perpendiculr to the plne contining AB nd AC. AB! AC = (8,4,4 nd hence ( 8,4,4 is vector perpendiculr to the plne. (,, eqully well serves s vector perpendiculr to the plne of course.

5 76 Exercise.. Find (,,4 " (!,,5. Find vector perpendiculr to oth (!,, nd (,,.. Find re of prllelogrm which hs (0,0,0 (,,5 (-,4,- nd (-,7,4 s its vertices. 4. Find re of! ABCwhere A is (,, B is (5,-, nd C is (7,0,6 5. Show tht! = "(! 6. Find vector perpendiculr to oth i! j nd j - k. 7.! i = (0,, nd i =. Find. 8. Find ngle etween (,,0 nd (,-, (0,0,0 (,, nd (,,n re three vertices of tringle whose re is. Find n. 0. True or Flse? (Assume nd re non-zero vectors in the hypotheses.! = 0 implies is sclr multiple of.! = implies =. c! = implies = 0 d i! i = k! k e! i =! i implies i = i f! i =! i implies k = k Exercise. Answers. ( 7,! 4,7. (,! 7, (,6, 7. (,!, n = 0 or.6 0. True Flse c True d True e Flse f True

6 77 Theorem. Let,, c e three non-coplnr vectors considered s djcent edges of prllelepiped s shown in digrm. c Then the volume of the prllelepiped is (! c Proof A B D O c C Let D e point in the se of the prllelepiped such tht AD is vector perpendiculr to the plne in which nd c lie. Then (! c =! (!! c! cos" (Theorem. where! is the ngle etween nd (! c. i.e.! = "OADin the digrm.

7 78 But note cos! = AD = height of prllelepiped where (! c is the re of the se. (Theorem. corollry. " (! c = re of se times height of prllelepiped. = volume of prllelepiped. The solute vlue symol out (! c in the sttement of the theorem cters for the possily tht! is otuse. This is n extremely importnt theorem for its corollry.,, c re coplnr iff (! c = 0 This follows immeditely from the fct tht if (! c = 0 then (! c = 0 nd hence the volume of the corresponding prllelepiped is zero, i.e. the prllelepiped is squshed flt i.e. is merely plne. i.e.,, c re coplnr. From now on we shll use (! c = 0 s test for coplnrity of vectors,, c in R. Note furthermore from the proof of theorem. tht c! nd c! eqully well re expressions for the volume of the prllelepiped. (! c is clled the triple sclr product of!,!, c! Exmple Find m so tht A (,,, B (,,, C (,4,5, D (,,m re coplnr points. AB = (,,, AC = (,,4, AD = (0,, m! But A,B,C,D coplnr implies tht AB, AC, AD re coplnr nd so AB AC! AD = 0 i.e. (,, (,,4 " (0,, m! = 0 i.e. (,, (m!,! m,4 = 0

8 79 i.e. m 5 = 0 i.e. m = 5 Note tht this is much etter nd quicker wy of hndling coplnrity questions then previously developed. Note lso tht since = nd! = "(! then! c =! c = c! etc. (see note fter Theorem. Exmple To find re of tringle with vertices A (,, B (, nd C (c,c C (c,c A (, (, B We hve estlished formul for re of tringle in R nd to find re of tringle ABC in question we merely emed (,,0 (c,c,0 respectively.! ABC in R y letting A,B,C hve co-ordintes (,,0 Clerly this produces tringle congruent to! ABC.

9 80 Then re of! ABC = AB! AC = (!,!, 0 " (c!, c!, 0 = c c! c!! c A mnemonic for this re is dding products nd c c sutrcting the products. Note tht this formul cn e pplied to ny polygon in R y simply listing the vertices in the proper sequence. Exmple To show: # ( # c " ( c! ( c Note tht!(! c is perpendiculr to (! c nd hence lies in the plne of nd c. This mens tht!(! c cn e represented s liner comintion of nd c which, note, ( c! ( c is. The proof of this identity is rther lengthy ut involves no more thn the mechnicl pplictions in computing!(! c nd is hence left s n exercise.

10 8 Exmple To show tht the volume of tetrhedron, three of whose concurrent edges re,, c, is (! c 6 A B O c D C Let the tetrhedron e s in the digrm. Its volume is the re of the se times the height (since tetrhedron is pyrmid. i.e. the volume of OABC =! (re " OBC! (height AD = (! c ( cos" =!c cos" 6 But note! is the ngle etween nd (! c since AD is perpendiculr to! OBC nd hence is multiple of! c.!volume of OABC = (! c using theorem.. 6

11 8 Exmple Consider tringle ABC with medins AM, BN, CP s shown B M P C N A B Let AB =, BC = nd CA = c A c C Now medins AM, BN, CP re, c nd c respectively. Note tht AM BN CP = ( ( c (c = ( c = 0 = 0 It follows tht the medins AM, BN, CP without rottions or mgnifiction, cn e individully trnslted to form tringle. Cll this tringle Δ RST. To find re of Δ RST formed y medins S c R c T

12 8 Now Are!RST = ( " #(c = ( " #(# # = ( " ( = " " " " = 0 " # ( " 0 = = 4 ( " 4 ( " = 4 Are!ABC Therefore the medins form tringle whose re is 4 of the re of the originl tringle. Exercise.. Find (, ",! (4,,. Find unit vector perpendiculr to oth (!,5, nd (,4,. Explin why! c is well defined, i.e. unmiguous. 4. True or Flse? Assume vectors re non-zero. i i! j = k ii i! k = j iii (! c = 0,, c re coplnr vectors iv! = 0 v! =! c = c vi = m nc! c = 0

13 84 vii viii =! the ngle etween nd is 45 or 5.! c = c! ix! c = c! x! (! c = (!! c xi (!! = 0 xii! =! xiii (! c = (! c (! c xiv! = 0 = m for some sclr m. xv xvi n! =! n =! iff = xvii "! xviii tn! = " where! is the ngle etween nd. xix " = (! " (! xx (! = (! xxi! =! = { } is coplnr set of vectors. xxii,,(!! c xxiii If,,c re not coplnr then! c > 0 xxiv "! xxv! = = 0 xxvi! i =! i nd! k =! k = xxvii It is possile for! =? xxviii!!! =! nd!! = 0!! =

14 85 5. Find the re of prllelogrm which hs (0,0,0 (-,4, (,-4, nd (0,0,4 s its vertices. 6. Find m so tht (,, (,,0, (0,, nd (m,m,0 re coplnr points 7. i Find re of! ABC where A is (,-, B is (0,, nd C is (,0,-. ii Try to find generl expression for re of nd C is (0,0,c.! ABC where A is (,0,0 B is (0,,0 8. Stte whether the following expressions re V vector, S sclr or M meningless. i (! (! ii ( c d iii (! (c! d iv (! c ( c v ( c 9. Find ngle etween (,,0 nd (,-,0 0. The vertices of tetrhedron re (0,0,0 (,-,0 (0,, nd (,,-. Find its volume.. is vector mking equl ngles with ech of the three xes in R. Find tht ngle.. Show tht if ny one of the vectors!,!, c! is liner comintion of the other two vectors then! c = 0

15 86 Hrder Questions. A D O B Given terms of! OAB s in the digrm, with OA = nd OB = nd., find the ltitude OD in 4. Find circumcentre of! ABCif A is (0,,4, B is (,,4 nd C is (0,,. Hint: i circumcentre is coplnr with A,B,C. ii circumcentre is the intersection of the perpendiculr isectors of sides of the tringle. 5. Show! (! c! (c! c! (! = 0 6. Show! = 7. Let A e (,0,0 B e (0,,0 nd C e (0,0, Find point P coplnr with A,B,C so tht OP is perpendiculr to the plne contining A,B,C,P. 8. Find the distnce from D to the plne contining A,B,C where A is (,, B is (,,, C is (,, nd D is (,, 9. Find the co-ordintes of the centre of grvity of the tetrhedron ABCD if A is (,,, B is (,,4, C is (-,,- nd D is (,4,7. 0. ABCD is squre. A is (,. C, the centre of grvity of the squre is (,5. Find the co-ordintes of B,C nd D.

16 87. Is (! c perpendiculr to?. If! =! c does it follow tht! c = 0?. In the twelve possile rrngements of! c,! c,c! etc. find those which re equl nd those which re negtives of ech other. Exercise. Answers. (! 8,6,. (,!, Becuse (! c mkes no sense the only wy to red! c is (! c. 4. i True ii Flse iii True iv True v Flse vi True vii True viii True ix Flse x Flse xi Flse xii True xiii True xiv True xv True xvi Flse xvii True xviii True xix Flse xix Flse xx True xxi Flse xxii True xxiii True xxiv Flse xxv True xxvi True xxvii True m = xxviii True 7. i 6 ii c c. 8. i S ii S iiis ivm v M rccos ( " " (!. (!

17 88 Note tht this vector is perpendiculr to (! nd hence liner comintion of nd, s required, nd perpendiculr to ( - the se BA (,, (,, (,, (0,7 (5,8 (6,. No. Yes

18 89 Prctice Test on Chpter. True or Flse?! = 0 implies = m for some sclr m. (! c = 0 implies = m nc. c The length of projection of j onto i k is 0. d! = implies = e! = c implies c = 0 f! = c implies c = 0 g i i = k k h i! (j! k = (i! j! k i! (! c = (!! c j! =! implies = k! =! implies = m for some sclr m.. Find the re of tringle of! ABCwhere A is (,, B is (,,4 nd C is (4,,.. Find the shortest distnce from (,, to the plne contining the points (,, (,, nd (,, 4. Descrie the distnce from point A to the line pssing through B nd C in terms of AC,BC,AC, etc. using dot product, cross product. 5. If nd c re known, does! = c lwys hve solution? Explin. Solve " (,, = (4,5,! 6. Decide whether it is true tht m! = m(! Decide whether it is true tht if! = c! then = mc for some sclr m. Prctice Test on Chpter Answers. T T ct df et ft gt ht if jf kt. 6

19 AB! AC BC 5. No ecuse nd c might not e perpendiculr to ech other. No solution. 6. Yes Flse Prctice Test on Chpter. Find the re of tringle ABC where A is (,, B is (,,5 nd C is (4,,6. Let {,,c} e coplnr set of vectors. Descrie wht vector (!! (! c might e. For exmple is this vector liner comintion of nd c, multiple of, the zero vector, vector perpendiculr to ech of,, c?. The line pssing through (,, nd (,,4 is prllel to the line pssing through (0,0,0 nd (,,. Find the shortest distnce etween these lines. 4. Find the shortest distnce from the origin (0,0,0 to the plne contining the points (,0,0 (0,,0 nd (0,0, 5. Find co-ordintes of point D so tht OD is perpendiculr to the plne contining the points (,0,0 (0,, nd (0,0, where O is the origin (0,0,0 6. Simplify (!! (! c if = c. 7. Is it possile for to equl!? If so, give n exmple. If not, explin why not. Prctice Test on Chpter Answers. 6. the zero vector. the zero vector (,0, 7. Yes, if the ngle etween nd is 45 or 5.

20 9 Prctice Test on Chpter. True or Flse?! = = c! =! d! =! implies = e If! = 0 then, re colliner f If (! is perpendiculr to c then c = m n for some vlues of m nd n. g v! (,, = 0 implies v = m(,, for some vlue of m. h! = c implies c = 0 i! i =! i implies i = i j! (! c = (!! c k! (! c = m nc for some vlues of m nd n. APFT is prllelogrm. A is (,, P is (,,5 F is (4,-, Find the co-ordintes of point T. Find the re of APFT. Find the vlue(s of n so tht (,, (,, (,, nd (0,0,n re coplnr. 4. Investigte whether you think it is true tht 5. Find vector so tht ( i! j = i (! c =! c! c 6. Let P e point so tht OP " (,, = (7,! 5, Find possile position for P Descrie in your own words the set of ll points P tht stisfy the eqution elow. 7. Find vector v so tht (,, " v = (,!,

21 9 nd (,, v = 6 8. Prove tht! = Prctice Test on Chpter Answers. F T c T d F e T f T g T h T i F j F k T. T is (,-,0 Are of APFT is. n = 4 4. True 5. ( -,,- where cn e ny numer 6. P cn e ny point of the form (t, t, t 5 7. v = (,,

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