Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 44In this formula we call (v·e 1 )e 1 and (v·e 2 )e 2 the vector components of v along e 1 ande 2 , respectively; and we call v · e 1 and v · e 2 the scalar components of v along e 1 ande 2 , respectively.If θ denote the angle between v and e 1 , thenv·e 1 = ‖v‖cosθ and v·e 2 = ‖v‖sinθand the decomposition (3.7) can be expressed asExample 3.16 Letv = 〈2,3〉, e 1 =v = (‖v‖cosθ)e 1 +(‖v‖sinθ)e 2 (3.9)〈 1 √2 ,〉 〈1√ , and e 2 = −√ 1 ,2 2〉1√2Find the scalar components of v along e 1 and e 2 and the vector components of v along e 1and e 2 .Solution .........Example 3.17 A rope is attached to a 100-lb block on a ramp that is inclined at an angleof 30 ◦ with the ground.30 ◦How much force does the block exert against the ramp, and how much force must be appliedto the rope in a direction parallel to the ramp to prevent the block from sliding down theramp?Solution .........Orthogonal ProjectionsThevectorcomponentsofv alonge 1 ande 2 in(3.8)arealsocalledtheorthogonal projectionsof v on e 1 and e 2 and are denoted byproj e1v = (v·e 1 )e 1 and proj e2v = (v·e 2 )e 2In general, if e is a unit vector, then we define the orthogonal projection of v on e tobeproj e v = (v·e)e (3.10)