Slide 1

Assigned work: pg.398 #1,3,7,8,11-14 pg. 419 #3 (work) Recall: Dot Product Slide 2 7.5 Scalar & Vector Projections Projections: The vector is the Vector Projection of on The scalar is the Scalar Projection of on O N A B Slide 3 7.5 Scalar & Vector Projections From the diagram derive the: Formula for the scalar projection of on : Also Recall: Dot So, O N A B Slide 4 7.5 Scalar & Vector Projections Formula for the scalar projection of on : Note: Scalar projection of on = Scalar projection of on L.S.R.S. Slide 5 7.5 Scalar & Vector Projections Finding Direction Angles of a Vector in R 3 : If is a unit vector, then the scalar projection of is just If then the scalar projection on the standard basis vectors are a,b,c. The vector projection on the axes are Slide 6 7.5 Scalar & Vector Projections Let be the angle makes with the positive x axis, positive y axis and positive z axis respectively. Then: Slide 7 7.5 Scalar & Vector Projections Therefore the direction angles that makes with the positive x,y and z axes respectively are: Slide 8 7.5 Scalar & Vector Projections Deriving the Vector Projection: Recall: Scalar projection is Multiply it by a unit vector pointing along and we will have: Vector Projection of O N A B Slide 9 7.5 Scalar & Vector Projections Deriving the Vector Projection (of a on b): unit vector pointing along O N A B Slide 10 7.5 Scalar & Vector Projections Ex 1: If find the scalar and vector projection of Slide 11 7.5 Scalar & Vector Projections Ex 2: Find the angle that vectormakes with the z-axis. Slide 12 7.5 Scalar & Vector Projections Ex 2: If find the scalar and vector projection of Slide 13 7.5 Scalar & Vector Projections Work: A force acting on an object does work when the object moves. The work done by a force is a scalar quantity equal to the product of the distance the object is displaced and the component of the force in the direction of the displacement. Slide 14 7.5 Scalar & Vector Projections A force,, moves a particle from O to A Its displacement vector is The Scalar projection of is Slide 15 7.5 Scalar & Vector Projections The work done by a force, in displacing an object through is: Force is measured in Newtons, N Distance is measured in metres, m Work is measured in Newton-metres (Nm) = Joules (J) Slide 16 7.5 Scalar & Vector Projections Ex 3: Calculate the work done by a 10.N force, in moving a particle from A(2,1) to B(8,5), when the force acts at an angle of 45 degrees to and the distance is measured in metres.