11.3 The Dot Product of Two Vectors

1. 11.3 The Dot Product of Two Vectors

2. Definitions Thedot product of u and vin the plane is (Read “u dot v”) The dot product of u and vin 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)

3. Examples

4. Properties Another form of the Dot Product:

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

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

7. Vector Components u w2 w1 v • Let u and v be nonzero vectors. • w1 is called the vector component of u alongv • (or projection of u onto v), and is denoted by projvu • w2 is called the vector component of u orthogonal tov

8. Application A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o 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

9. A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o 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 u

10. A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o 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 v 60o E u

11. A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o 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 We need to find the magnitude and direction of the resultant vectoru + v. v u+v E u

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

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

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