# Handwritten i denotes a unit vector examples 1 i i is

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^ ^ ^ ( Handwritten : I denotes a unit vector ) . Examples : 1) I = I is a unit vector Note I = tin to I t ok . Hence I I I = 1 . 2) I = 2in - 31 tk . Since I II = 2 't f - si ' th = Fi ; k is not a unit vector . However ÷ , = ÷ , = in - faint if , " check I is a unit vector ! I II tf Y is a 2b vector then the direction of I can be worked out from the angle o that is formed with the X - axis .
L14 - Vectors - Direction cosines School of Mathematical Sciences Page 4 ENG1090 3. Direction angles and direction cosines The direction angles of a vector are the angles that the vector makes with each coordinate axis, ie the directions of i , j , and k . By convention, these angles are measured in a positive sense (only) from each coordinate axis. It is sometimes more convenient to identify the cosines of these angles, called the direction cosines . In two-dimensional space, the direction cosines for the vector a b ± v i j are given by cos a D v for the direction relative to i and cos b E v for the direction relative to j and the corresponding direction angles are therefore obtainable from evaluating Arccos ( ) | | a D v and Arccos( ) | | b E v . Note that for any vector these angles will always be in the ranges 0 D S d d and 0 E S d d , so the Arccos (principal) version of the inverse cosine function with that range (as shown in the diagram in lecture 4) must be used. v y j i x a b β x y bj a- - - - - - - - - - - - ; l l l l l ai ? x IV Analysis : ya : Cosa = a- for the direction relative to I < I a' in ' t " ( we measure a with respect to ~ a - - Arcos ( Ii ) the positive x-axis ) bin A- - - - < i # , Corrs = b- for the direction relative to I IVI ~ ( we measure s with respect to the positive v a- Areca - ( Iv , ) Y - axis ) .