Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 45The orthogonal projection of v on an arbitrary nonzero vector b can be obtained by normalizingb and then applying Formula (3.10); that is,( )( )b bproj b v = v·‖b‖ ‖b‖which can be rewritten asproj b v = v·b‖b‖ 2 b (3.11)Moreover, if we subtract proj b v from v, then the resulting vectorv−proj b vwill be orthogonal to b; we call this the vector component of v orthogonal to b.v−proj b vvvv−proj b vbproj b vproj b vbAcute angle between v and bObtuse angle between v and bExample 3.18 Find the orthogonal projection of v = i +j+k on b = 2i+2j, and thenfind the vector component of v orthogonal to b.Solution .........WorkRecall that we define the work W done on the object by a constant force of magnitude Facting in the direction of motion over the distance d to beW = Fd = force × distance (3.12)If we let F denote a force vector of magnitude ‖F‖ = F acting in the direction of motion,then we can write (3.12) asW = ‖F‖dMoreover, if we assume that the object moves along a line from point P to point Q, thend = ‖ −→ PQ‖, so that the work can be expressed entirely in vector form asW = ‖F‖‖ −→ PQ‖The vector −→ PQ is called the displacement vector for the object.In the case where a constant force F is not in the direction of motion, but rather makesan angle θ with the displacement vector, then we define the work W done by F to beW = (‖F‖cosθ)‖ −→ PQ‖ = F·−→ PQ (3.13)