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 of sclrs re distnce nd speed. Representtion of Vectors Column vector form æ In column vector form ç b the represents the movement in the positive x direction nd the b movement in the positive y direction. Vectors cn lso be represented using lower cse bold letter. For exmple we could use to represent the vector. 3 4 Unit vector form. æ We cn write ç b s i + bj where i nd j re vectors of length 1 unit in the directions of x nd y respectively. i nd j re clled bse vectors. j i When considering objects tht move round in three dimensionl spce we represent vector in similr wy s bove but introduce the letter k for the vector of length 1 unit in the z direction.
k j i Equl vectors Two vectors re equl if they hve the sme direction nd the sme mgnitude Q P It does not mtter where in the Crtesin plne these vectors re they re still prllel. These vectors re free vectors nd cn be nywhere in 3D spce. Vectors nd PQ re pointing in the sme direction (re prllel) nd hve the sme mgnitude. PQ If two vectors re equl in length then their components will be the sme. Negtive vectors The two vectors nd MN hve the sme mgnitude but re prllel nd in opposite directions. So MN. MN. MN is clled the negtive vector where M N The direction of vector is importnt, not just its length.
We cn write s Prllel Vectors Vectors,CD nd EF re ll prllel but hve different mgnitudes. D E F C Two vectors re prllel if one is sclr multiple of the other. i.e. if krs where k is sclr quntity. In the lterntive nottion this could lso be written s kb Vectors nd GH hve the sme mgnitudes but different directions. GH H G Position vectors Position vectors re vectors giving the position of point, reltive to fixed origin, O.
Resultnt vectors O The resultnt vector = O - O Therefore to find the resultnt vector between two points nd we cn subtrct the position vector of from the position vector of Similrly if we know vector PQ nd the vector PR then ech of the points Q nd R re given reltive to point P. Q R P Now QR QP PR PR PQ Colliner Points Colliner points ll lie in stright line
The Mgnitude of Vector The mgnitude of is the length of the vector nd is denoted by. Other nmes for mgnitude re modulus, length, norm nd size. Mgnitude is found by using Pythgors theorem, 2 2 If i bj then b. b In three dimensions this cn be extended. 2 2 2 If b i bj ck then b c c The distnce between two points in spce b - b O If ( x, y, z ) then O x i y j z k 1 1 1 1 1 1 nd if ( x, y, z ) then b O x i y j z k O O O O b 2 2 2 2 2 2 ( x x ) i ( y y ) j ( z z ) k 2 1 2 1 2 1 Distnce ( x x ) ( y y ) ( z 2 2 2 1 2 1 z ) 2 2 1 Unit vectors
Unit vector is vector of length 1. vector of length 1 in the direction of cn be found using the formul vector of length k in the direction of cn be found using the formul k ddition nd Subtrction of vectors ddition of vectors u v Now u + v is interpreted geometriclly s first move long vector u followed by move long vector v. The resultnt vector, u + v, is the third side of the tringle formed when u nd v re plced next to ech other tip to toe.
u+v v u Notice lso tht vector ddition is commuttive since u + v = v + u. This gives rise to the prllelogrm of vector ddition. u vu+v v u Notice tht ddition nd subtrction cn esily be found rithmeticlly by dding or subtrction the corresponding components æ u = ç b æ v = ç c d æ u + v = ç æ u - v = ç b b + æ c ç d - æ c ç d Subtrction of vectors = æ + c ç b + d = æ - c ç b - d
u v = u + (-v) u v is interpreted geometriclly s move long vector u followed by move long negtive v or u + ( v) The Zero Vector Consider this tringle Q P R PQ QR RP must be equl to zero s the overll journey results in return to the strting point. This is written s PQ QR RP 0 The zero is in bold type to indicte it is vector 0 0 0 in two dimensions nd 0 in three. 0 0