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Dot Product of Two Vectors a, b c, d ac+bd Example 1 4,5 2,3 Example 2 u= 1,3 (u v ) w v= 2,4 w= 1,2 u 2v Angle Between Two Vectors a b cos a b Example 3 a b cos a b Find the angle between the two vectors 3,4 and 1,5 a b Use the formula cos a b to determine the the angle between two vectors whose dot product is zero. ~Flashback~ A=30º a=12 B=45º b= C= c= This is an oblique triangle with what given information? What do you use to solve for the sides? What is angle C and sides b & c ? 6.4 Finding Vector Components & Work Added Two vectors projv u ( uv v 2 )v Find the projection of u=<3,5> onto v=<6,2>. Write u as the sum of two orthogonal vectors, one of which is projvu proj u ( u v )v v v 2 1500 lb. boat at a 20 degree incline. projv u ( uv v 2 )v 200 lb. cart on a 30 degree incline. projv u ( uv v 2 )v W F DWork A force of 45 lbs. in the direction of 30 degrees above the horizontal is required to slide a table across the floor. Find the work done if the table is dragged 20 feet. W FD 50 lb. force, 12 ft. wide door 58) A force of 45 pounds in the direction of 30 degrees above the horizontal is required to slide a table across a floor. Find the work done if the table is dragged 20 feet. 59) A toy wagon is pulled by exerting a force of 20 pounds on a handle that makes a 25 degree angle with the horizontal. Find the work done in pulling the wagon 20 feet. 60) A mover exerts a horizontal force of 25 pounds on a crate as it is pushed up a ramp that is 12 feet long and inclined at an angle of 20 degrees above the horizontal. Find the work done on the crate.