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 ab±vijare given by cosaDvfor the direction relative to iand cosbEvfor the direction relative to jand the corresponding direction angles are therefore obtainable from evaluating Arccos ()||aDvand Arccos()||bEv. Note that for any vector these angles will always be in the ranges 0DSddand 0ESdd, 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 βx y bja------------;lllllai?xIVAnalysis:ya:Cosa=a-forthedirectionrelativetoI<Ia'in't"(wemeasureawithrespectto~a--Arcos(Ii)thepositivex-axis)binA----㱺<i#,Corrs=b-forthedirectionrelativetoIIVI~(wemeasureswithrespecttothepositiveva-Areca-(Iv,)Y-axis).