Math 151 Section 1 2 The Dot Product Work The work W done by a constant force F in moving an object through a distance d is given by W Fd This formula only applies when the force is directed along the line of motion of the object Suppose we have an object moving from point P to point Q under a force F as shown in the figure F P Q S The work done in moving the object from P to Q depends on two things 1 The distance the object has been moved given by D PQ where D is the displacement vector 2 The magnitude of the force applied in the direction of D given by PS F cos So the work done by moving the object is given by W D F cos Example Find the work done by a force of 10 N acting in the direction N50 W in moving an object 4 m due west The Dot Product The dot product of vectors a and b is given by a b a b cos where is the angle between a and b and 0 If a 0 or b 0 then a b 0 Example Prove a b a b 0 Math 151 The Dot Product Version 2 The dot product of vectors a a1 a2 and b b1 b2 is given by a b a1b1 a2 b2 Properties of the Dot Product are given on p 57 of the textbook Example Calculate the dot product of 4i and 3j Example A force given by the vector F 3 8 moves an object along a straight line from point 2 3 to point 4 5 Calculate the work done if the distance is measured in meters and the magnitude of the force is measured in Newtons Example Determine whether the given vector pairs are parallel perpendicular or neither A 2 6 3 1 B 2 6 3 9 Math 151 Orthogonal Complement The orthogonal complement of a vector a a1 a2 is given by a a2 a1 Example Find a unit vector perpendicular to 2 3 Vector and Scalar Projections Given vectors a and b The scalar projection of b onto a is compa b The vector projection of b onto a is proja b a b a a b a 2 a Example Find the scalar and vector projections of 4 8 onto 2 1 Math 151 Example Find the distance from the point P 2 1 to the line y 2x 1
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