Vectors Practice Part B - MadAsMaths

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Created by T. MadasVECTORPRACTICEPart BCreated by T. Madas

Created by T. MadasTHECROSSPRODUCTCreated by T. Madas

Created by T. MadasQuestion 1Find in each of the following cases a b , where the vectors a and b area) a 2i 5 j k and b 3i jb) a i 2 j k and b 3i j kc) a 3i j 2k and b i 3 j kd) a 7i j 4k and b i 3 j 2ke) a 2i 5 j 4k and b 3i j 3ki 3 j 17k , i 4 j 7k , 5i 5 j 10k , 10i 18 j 22k , 11i 18 j 17kQuestion 2Find a unit vector perpendicular to botha 2i 5 j kandb 3i j k .1( 4i 5 j 17k )330Created by T. Madas

Created by T. MadasQuestion 3The vectors a and b , are not parallel.Simplify fully( 2a b ) ( a 2b ) .5b a 5a bQuestion 4The vectors a , b and c are not parallel.Simplify fullya i b ( c a ) .aCreated by T. Madasi(b c)

Created by T. MadasQuestion 5The following vectors are givena 2i 3 j kb i 2j kc j 3ka) Show that the three vectors are coplanar.b) Express a in terms of b and c .a 2b cCreated by T. Madas

Created by T. MadasQuestion 6The following three vectors are givena i 3 j 2kb 2i 3 j kc i 2 j λkwhere λ is a scalar constant.a) If the three vectors given above are coplanar, find the value of λ .b) Express a in terms of b and c .λ 1 , a 3c bCreated by T. Madas

Created by T. MadasQuestion 7The vectors a , b and c are such so thatc a iandb c 2k .Express ( a b ) ( a b 2c ) in terms of i and k . 2i 4kCreated by T. Madas

Created by T. MadasCROSS PRODUCTGEOMETRICAPPLICATIONSCreated by T. Madas

Created by T. MadasQuestion 1Find the area of the triangle with vertices at A (1, 1, 2 ) , B ( 1, 2,1) and C ( 2, 3,3) .1 32Question 2Find the area of the triangle with vertices at A ( 2,1,1) , B ( 1, 0, 4 ) and C ( 3, 1, 1) .1 1222Created by T. Madas

Created by T. MadasQuestion 3A triangle has vertices at A ( 2, 2,0 ) , B ( 6,8,6 ) and C ( 6,8,12 ) .Find the area of the triangle ABC .90Created by T. Madas

Created by T. MadasQuestion 4A parallelepiped has vertices at the points A ( 2,1, t ) , B ( 3,3, 2 ) , D ( 4,0,5 ) andE (1, 2,7 ) , where t is a scalar constant.HGEFDCAB a) Calculate AB AD , in terms of t . b) Find the value of AB AD i AEThe volume of the parallelepiped is 22 cubic units.c) Determine the possible values of t .(12 3t ) i ( t 1) j 5kCreated by T. Madas, 11t 44 , t 2,6

Created by T. MadasQuestion 5A triangular prism has vertices at the points A ( 3,3,3) , B (1,3, t ) , C ( 5,1,5) andF ( 8,0,10 ) , where t is a scalar constant.FCDAEBThe face ABC is parallel to the face DEF and the lines AD , BE and CF are parallelto each other. a) Calculate AB AC , in terms of t . b) Find the value of AB AC i AD , in terms of t .The value of t is taken to be 6 .c) Determine the volume of the prism for this value of t .d) Explain the geometrical significance if t 1 .( 2t 6 ) i ( 2t 2 ) j 4k, 4t 4 , V 14 cubic units ,A, B, C , D are coplanar, so no volumeCreated by T. Madas

Created by T. MadasQuestion 6A tetrahedron has vertices at the points A ( 3,6, 4 ) , B ( 0,11,0 ) , C ( 4,1,28 ) andD ( 7, k , 24 ) , where k is a scalar constant.a) Calculate the area of the triangle ABC .b) Find the volume of the tetrahedron ABCD , in terms of k .The volume of the tetrahedron is 150 cubic units.c) Determine the possible values of k .area 75 , volume 50 k 6 , k 3,153Created by T. Madas

Created by T. MadasQuestion 7With respect to a fixed origin O the points A , B and C , have respective coordinates( 6,10,10 ) , (11,14,13) and ( k ,8,6 ) , where k is a constant.a) Given that all the three points lie on a plane which contains the origin, find thevalue of k .b) Given instead that OA , OB , OC are edges of a parallelepiped of volume 150cubic units determine the possible values of k .k 10 , k 5, 25Created by T. Madas

Created by T. MadasLINESCreated by T. Madas

Created by T. MadasQuestion 1Find an equation of the straight line that passes through the point P (1, 4,1) and isparallel to the vector 3i 4 j k .Give the answer in the form r a b where a and b are constant vectors.r ( 3i 4 j k ) 8i 4 j 8kQuestion 2Find an equation of the straight line that passes through the points P ( 5,0,9 ) andQ ( 8, 4,10 ) .Give the answer in the form r a b where a and b are constant vectors.r ( 3i 4 j k ) 36i 22 j 20kCreated by T. Madas

Created by T. MadasQuestion 3A straight line has equationr 4i 2 j 5k λ ( i 8 j 3k ) ,where λ is a scalar constant.Convert the above equation into Cartesian form.x 4 y 2 z 5 18 3Question 4A straight line has equationr 2i 3j λ ( i 2 j 2k )where λ is a scalar parameter.Convert the above equation into Cartesian form.x 2 y 3 z 1 22Created by T. Madas

Created by T. MadasQuestion 5Convert the equation of the straight linex 3 y 2 5 z 237into a vector equation of the form r a λ b , where a and b are constant vectors andλ is a scalar parameter.r 3i 2 j 5k λ ( 2i 3 j 7k )Question 6A straight line has equation r ( 5i 2 j 3k ) ( 7i 5k ) 0Convert the above equation into the form r a λ b , where a and b are constantvectors and λ is a scalar parameter.x 5 z 3 , y 275Created by T. Madas

Created by T. MadasQuestion 7A straight line has equationr ( 2i 4 j 3k ) ( 2i 5 j 8k )Convert the above equation into the form r a λ b , where a and b are constantvectors and λ is a scalar parameter.r 3i 2 j 2k λ ( 2i 4 j 3k ) or r i 2 j k λ ( 2i 4 j 3k )Question 8If the point A ( p, q,1) lies on the straight line with vector equationr ( 2i j 3k ) ( 8i 7 j 3k ) ,find the value of each of the scalar constants p and q .p q 3Created by T. Madas

Created by T. MadasQuestion 9The straight line L has equation r ( 3i 2 j 3k ) ( 2i 3k 4k ) 0 .Use a method involving the cross product to show that the shortest distance of the point( 2, 1, 3) from L is 3 units.proofCreated by T. Madas

Created by T. MadasPLANESCreated by T. Madas

Created by T. MadasQuestion 1Find a Cartesian equation of the plane that passes through the point A ( 6, 2,5) , and itsnormal is in the direction 5i 2 j 3k .5 x 2 y 3 z 49Question 2Find a Cartesian equation of the plane that passes through the point A ( 5,1, 2 ) , and itsnormal is in the direction 2i 7 j k .2x 7 y z 5Created by T. Madas

Created by T. MadasQuestion 3Find a Cartesian equation of the plane that passes through the pointsA ( 5, 2, 2 ) , B ( 1, 2,1) andC ( 3, 2, 2 ) .2 x 11 y 12 z 8Question 4Determine a Cartesian equation of the plane that contains the point A ( 9, 1, 0 ) and thestraight line with vector equationr 5i 2 j 2k λ ( i 3j 6k ) ,where λ is a scalar parameter.24 x 26 y 9 z 190Created by T. Madas

Created by T. MadasQuestion 5Find a Cartesian equation of the plane that contains the parallel straight lines withvector equationsr1 2i j 5k λ ( i j k )and r2 3i j 6k µ ( i j k ) ,where λ and µ are scalar parameters.z x 3Question 6Find a Cartesian equation of the plane that contains the intersecting straight lines withvector equationsr1 6i 3j 7k λ ( i j 2k ) andr2 6i 2 j 4k µ ( 2i j k ) ,where λ and µ are scalar parameters.x 3y z 4Created by T. Madas

Created by T. MadasQuestion 7a) Find a set of parametric equations for the plane that passes through the pointsA ( 2, 4,1) , B ( 6, 0, 2 ) and C ( 0,1,7 ) .b) Eliminate the parameters to obtain a Cartesian equation of the plane.( x, y, z ) ( 2 2λ 4µ , 4 3λ 4µ ,1 6λ 3µ ) ,Created by T. Madas33 x 18 y 20 z 158

Created by T. MadasGEOMETRICPROBLEMS(WITH PLANES AND LINES)Created by T. Madas

Created by T. MadasQuestion 1Find the coordinates of the point of intersection of the plane with equationx 2 y 3z 4and the straight line with equationr i j 5k λ ( i j 2k ) ,where λ is a scalar parameter.(1,3, 1)Question 2Find the coordinates of the point of intersection of the plane with equation3x 2 y 7 z 2and the straight line with equationr 9i 2 j 7k λ ( i 3j 2k ) ,where λ is a scalar parameter.( 5, 10, 1)Created by T. Madas

Created by T. MadasQuestion 3Find the size of the acute angle formed by the planes with Cartesian equations4 x 4 y 7 z 13 and 7 x 4 y 4 z 6 .78.6 Question 4Find the size of the acute angle between the planes with Cartesian equations4 x 5 y 3 z 82 and 2 x 5 y 6 z 124 .52.1 Created by T. Madas

Created by T. MadasQuestion 5Find the size of the acute angle between the plane with equation2 x 2 y z 12and the straight line with equationr 7i j 2k λ ( 2i 6 j 3k ) ,where λ is a scalar parameter.31.6 Question 6Find the size of the acute angle formed between the plane with Cartesian equation2x 2 y z 2and the straight line with vector equationr 2i j 5k λ ( 3i 4 j 12k ) ,where λ is a scalar parameter.14.9 Created by T. Madas

Created by T. MadasQuestion 7Find the size of the acute angle between the plane with equation3x 2 y z 5and the straight line with equationr 3i 2 j 3k λ ( 2i 3j 4k ) ,where λ is a scalar parameter.52.6 Question 8Find shortest distance of the origin O from the plane with equation4 x 3 y 5 z 20 .2 2Created by T. Madas

Created by T. MadasQuestion 9Find shortest distance of the origin O from the plane with equationx 2 y 2z 5 .53Question 10Find shortest distance from the point A (1,3, 2 ) to the plane with Cartesian equationx 3 y 5z 5 .1535Created by T. Madas

Created by T. MadasQuestion 11Find shortest distance from the point P ( 3,1,3) to the plane with Cartesian equationx y 2z 2 .6Question 12Find shortest distance of the point P (1, 2,9 ) from the plane with Cartesian equation x 4 y 8 z 16 .7Created by T. Madas

Created by T. MadasQuestion 13Find the distance between the parallel planes with Cartesian equations2 x 6 y 3 z 70and2 x 6 y 3 z 14 .8Question 14The straight line with vector equationr ( λ 5) i ( 2 λ ) j ( λ 2 ) k ,where λ is a scalar parameter, is parallel to the plane with Cartesian equationx 2 y z 10 .Find the distance between the plane and the straight line.66Created by T. Madas

Created by T. MadasQuestion 15Find the distance between the parallel planes with Cartesian equations3 x 2 y z 20and3 x 2 y z 40 .10 147Question 16Find the distance between the parallel straight lines with vector equationsr1 i 2 j k λ ( 5i 4 j 3k ) andr2 2i k µ ( 5i 4 j 3k ) ,where λ and µ are a scalar parameters.21 210Created by T. Madas

Created by T. MadasQuestion 17Find the distance between the parallel straight lines with vector equations r ( 2i k ) ( i j k ) 0andx 4 y 8 z 7 .2 6Question 18Find the shortest distance between the skew straight lines with vector equationsr1 i λ ( j k ) andr2 i 3j k µ ( 2i j k ) ,where λ and µ are a scalar parameters.2 2Created by T. Madas

Created by T. MadasQuestion 19Find the shortest distance between the skew straight lines with vector equationsr1 7i λ ( 7i 10k )andr2 3i 3 j k µ ( i 3j k ) ,where λ and µ are a scalar parameters.6 65Question 20Find the shortest distance between the skew straight lines with vector equationsr1 2i j k λ ( j 3k )andr2 i 2 j 3k µ ( i 2k ) ,where λ and µ are a scalar parameters.514Created by T. Madas

Created by T. MadasQuestion 21Find the intersection of the planes with Cartesian equations2 x 2 y z 2 and x 3 y z 5 ,giving the answer in the form r a λ b ,where a and b are constant vectors and λ is a scalar parameter.r i 2 j λ ( 5i 3 j 4k )Question 22Show that the planes with Cartesian equations4 x 5 y 3 z 82 and 2 x 5 y 6 z 124intersect along the straight line with equationr ( λ 6 ) i ( 20 2λ ) j ( 2λ 2 ) k ,where λ is a scalar parameter.proofCreated by T. Madas

Created by T. MadasQuestion 23The planes Π 1 and Π 2 have Cartesian equations:Π 1 : x 2 y 2z 0Π 2 : 3x 2 y z 5Show that the two planes intersect along the straight line with Cartesian equationx 4 y 3 z 1 .674proofCreated by T. Madas

The following three vectors are given 3 2 2 3 2 λ a i j k b i j k c i j k where λ is a scalar constant. a) If the three vectors given above are coplanar, find the value of λ. b) Express a in terms of b and c. λ 1 , a c b 3

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