1. C H. 6 – A DDITIONAL T OPICS IN T RIGONOMETRY 6.4 – Dot Products

2. M ULTIPLYING VECTORS There are 2 major ways to multiply vectors u v is the dot product of u and v u x v is the cross product of u and v (don’t worry about this now) Dot Product : The dot product of 2 vectors gives a scalar (non- vector) answer of 1 number Ex: If u = and v =, find u v. u v = (3)(4) + (6)(-1) = 12 – 6 = 6 Properties of the dot product: u (v w) = (u v) w u u = ||u|| 2

3. D OT P RODUCTS AND A NGLES Another formula for the dot product is: θ is the measure of the angle between the two vectors Ex: Find the measure of the angle between the vectors u = and v =. First find the dot product and the 2 magnitudes… …then solve for θ!

4. U SES FOR DOT PRODUCTS Orthogonal = perpendicular If 2 vectors are orthogonal, what is their dot product? Zero, because cos90 ° = 0 If 2 vectors are parallel, their slopes will be the same Ex: and are parallel vectors because they both have a slope of 5/3

5. F IND THE DOT PRODUCT OF U = 3 I -6 J AND V = 5 I + 2 J. 1. -8 2. 27 3. 3 4. 28 5. 10

6. U = AND V =. F IND THE MEASURE OF THE ANGLE BETWEEN U AND V. 1. 15.26 ° 2. 87.49° 3. 70.22° 4. 25.26° 5. 63.82°

7. U = AND V =. D ESCRIBE THE RELATIONSHIP BETWEEN U AND V. 1. Orthogonal 2. Parallel 3. Neither

8. U SES FOR DOT PRODUCTS : W ORK The work W done by a constant force F in the direction of a distance vector d is given by: Ex: To move a desk 19 m along the floor, you push the desk with a force of 12 N at a 30° angle from the ground. Find the amount of work you did. Draw a picture! Move the vectors tail to tail! Do the math! 12 N 19 m 30°