1. 11.3 The Dot Product of Two Vectors

2. The dot product of u and v in the plane is The dot product of u and v in space is Two vectors u and v are orthogonal  if they meet at a right angle.  if and only if u ∙ v = 0 (since slopes are opposite reciprocal) (Read “u dot v”) Definitions

3. Examples

4. Another form of the Dot Product: Properties

5. Find the angle between vectors u and v : Examples

6. Angles between a vector v and 3 unit vectors i, j and k are called direction angles of v, denoted by α, β, and γ respectively. Since we obtain the following 3 direction cosines of v: So any vector v has the normalized form: Direction Cosines

7. Let u and v be nonzero vectors.  w 1 is called the vector component of u along v (or projection of u onto v), and is denoted by proj v u  w 2 is called the vector component of u orthogonal to v w2w2 w1w1 u v Vector Components

8. A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60 o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N E Application

9. N E u

10. N E v u 60 o

11. A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60 o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N E v u We need to find the magnitude and direction of the resultant vector u + v. u+v

12. N E v u The component forms of u and v are: u+v 500 70 Therefore: and:

13. N E The new ground speed of the airplane is about 538.4 mph, and its new direction is about 6.5 o north of east. 538.4 6.5 o

14. 1) Compute 2) Compute 4) Find the angle between vectors v and w. 3) List pairs of orthogonal and/or parallel vectors. 6) Find the projection of w onto u. 5) Find the unit vector in the direction u. 7) Find vector component of w orthogonal to u. Examples