1. Assigned work: pg.398 #1,3,7,8,11-14 pg. 419 #3 (work) Recall: Dot Product

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

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

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.

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

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:

7. 7.5 Scalar & Vector Projections Therefore the direction angles that makes with the positive x,y and z axes respectively are:

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

9. 7.5 Scalar & Vector Projections Deriving the Vector Projection (of a on b): unit vector pointing along O N A B

10. 7.5 Scalar & Vector Projections Ex 1: If find the scalar and vector projection of

11. 7.5 Scalar & Vector Projections Ex 2: Find the angle that vectormakes with the z-axis.

12. 7.5 Scalar & Vector Projections Ex 2: If find the scalar and vector projection of

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.

14. 7.5 Scalar & Vector Projections A force,, moves a particle from O to A Its displacement vector is The Scalar projection of is

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)

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.