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. Vector B with mgnitude cm nd direction due north. b Vector MN with mgnitude cm nd direction south west. c Vector mgnitude cm nd direction due west. d Vector b mgnitude 8 cm nd direction 6. e Vector c mgnitude. cm nd direction. f Vector d mgnitude. cm nd direction. Write ech vector s column vector. B= ( ) becuse from to B is right down. Q C E F G B P D H b c e f d PQ = DC = EF = GH =............ = b = c =.......... d = e = f =......... On Resource sheet 8.b drw the following vectors. Lbel ech vector. B = b CD = c EF = d = e b = f c = g d = h MN =
8. Vectors nd vector nottion Using Resource sheet 8.b for ech pir of points below: i plot the points on the xes ii write down vector joining the two points. ( ) B(8 ) b C( ) D( ) c E( ) F( 8) d G( ) H( ) e I( ) J( 8) f K( 8) L(8 8) g M(8 ) N( 6) Plot the following points on Resource sheet 8.c. ( ) B( 6) C( 6) D( ) b Drw the vectors B DC nd D c Write ech vector s column vector. 6 Plot these points on the sme grid in Resource sheet 8.c. E( ) F( 6) G( ) H(6 ) b Write down the vectors i EF ii HG iii EH iv FG c Wht do you notice bout these vectors? d Wht kind of shpe is EFGH? 7 PQRS is squre. P is the point ( ). PQ = Plot P Q nd R on Resource sheet 8.c. b Mrk the vectors PQ nd PR. c Mrk the point S so tht PQRS is squre. d i Write down the vectors SR nd PQ. ii Wht do you notice bout these vectors? 8 These re the end points of three equl vectors. y 6 nd PR = 6 x Join up the pirs of points with equl vectors. b Write down the column vectors of these pirs of points.... B
8. The mgnitude of vector x Remember: the mgnitude (length) of the vector ( ) is x + y y Clculte the mgnitude of ech vector i s surd ii correct to deciml plce where necessry. Use the correct nottion to write your nswer e.g. = B = or B = J K = ( ) (i) J K = ( ) + = + 9 = (we could just write JK = ) (ii) J K =.8 ( d.p.) = b B = c PQ = d d = e e = f MN i... = g f = h DE 6 = ii... i... ii... i... ii... i... ii... i... ii... i... ii... i... ii... i... ii... Using Resource sheet 8. for ech pir of points: i plot the points on the xes ii write down vector joining the two points iii find the mgnitude of the vector correct to deciml plce. ( ) B(7 8) b C( 7) D( 6) c E( 7) F( 8) d G( 6) H(6 ) e I(7 ) J(7 ) f K( ) L( ) g M( ) N( ) 7
8. ddition of vectors Questions re trgeted t the grdes indicted The digrm shows some vectors. b c d e Drw the following vector sums on the squred pper below. Write ech vector sum s column vector. + d Join to d so tht their rrows point in the sme direction. + d = ( ) d + d + b... b b + d... c + c... d e + d... e c + e... f b + b... g + e + d... h + b + c... 9
8. ddition of vectors Work out the following vector sums. c e g + + + + Given tht = + b b = nd c = b d f h + + + + find s column vector b b + c c + c d + e + b + c f b + b + c g + c + c For ech pir of vectors B nd : i mke rough sketch of the vectors ii drw vector C iii write C s column vector. B = ( ) nd B C ( ) C C = ( ) + ( ) = ( ) B B = nd = 9B
8. ddition of vectors b B = nd = c B = nd = d B = nd = e B = nd = f B = nd = 9C
8. Prllel vectors Questions re trgeted t the grdes indicted The digrm shows some vectors. b c d e Drw the following vector sums on the squred pper below. Write ech vector sum s column vector. hs twice the mgnitude of nd is in the sme direction. b hs the sme mgnitude of b but is in the opposite direction. b = ( ) b = ( )... b b... c... d e... e... f d... g b... h c... 6
8. Prllel vectors Given tht = b = 6 nd c = x 8 b = x ( ) = ( ) = 6 x 6 ( ) find the following s column vector.... b b... c... d c... e b... f c... g + c... h + c... i b + c... j c... k b c... l c... m b... n c... o b + c... Find the pirs of prllel vectors. 9 6 6 ( ) is prllel to ( ) becuse ( ) = x ( ) 9............ 6B
8. Prllel vectors y 6 B * D C 6 x Drw the position vector of ech point on the digrm. Hint join the origin O to the point. b Write down the position vectors s column vectors............. c Complete the following: B = O + OB =... +... =... CD = CO +... =... +... =... d Wht cn you sy bout the lines B nd CD?...... 6C
8. Solving geometric problems in two dimensions Questions re trgeted t the grdes indicted From the given informtion decide whether points B nd C lie on stright line. 6 B = ( ) B C = ( ) B nd C will lie on stright line if B nd B C hve the sme direction. 6 ( ) = ( ) nd so B nd B C re prllel. lso they meet t B. B nd C lie on stright line. * B = =... b B = = 6... c B = = 6... d B = = 6 6... e B = =... f B = =... g B = 6 =... h B = =... From the given informtion decide whether points P Q nd R lie on stright line. P Q = + b P R = + 6b P Q nd R will lie on stright line if P Q nd P R hve the sme direction. P Q = ( + b) = + b which is not prllel to P R = + 6b. P Q nd R do not lie on stright line. PQ = + b PR = 6 + 6b.... b PQ = b PR = 6b...... c PQ = + b PR = 9 + b...... 6
8. Solving geometric problems in two dimensions d PQ = b PR = 6 + 9b... *... e PQ = + 6b PR = 6 + 9b...... f PQ = + b PR = b...... g PQ = b PR = + b...... h PQ = + b PR = + b.... The digrm shows the position vectors of points B nd C from O. Find i B... ii... b Show tht B nd C lie on the sme stright line. + b B + b...... O C In the vector digrm DC = B. Find i DC... ii D... Line is extended to point E so tht CE =. B 6b C b Find i CE... D E ii DE... c Wht cn you sy bout points D nd E? Give resons for your nswers.......... 6B