The Dot Product Angles Between Vectors Orthogonal Vectors

1. The Dot Product Angles Between Vectors Orthogonal Vectors The beginning of Section 6.2a

2. Definition: Dot Product The dot product or inner product of u = u , u and v = v , v is 1 2 1 2 u v = u v + u v 1 1 2 2 vector! The sum of two vectors is a… vector! The product of a scalar and a vector is a… scalar! The dot product of two vectors is a…

3. Properties of the Dot Product Let u, v, and w be vectors and let c be a scalar. 1. u v = v u 2 2. u u = |u| 3. 0 u = 0 4. u (v + w) = uv + uw (u + v) w = uw + vw 5. (cu) v = u (cv) = c(uv)

4. Finding the Angle Between Two Vectors v – u v u 0

5. Theorem: Angle Between Two Vectors v – u If 0 is the angle between nonzero vectors u and v, then v u 0 and

6. Definition: Orthogonal Vectors The vectors u and v are orthogonal if and only if u v = 0. • The terms “orthogonal” and “perpendicular” are nearly synonymous (with the exception of the zero vector)

7. Guided Practice Find each dot product = 23 1. 3, 4 5, 2 = –10 2. 1, –2 –4, 3 3. (2i – j) (3i – 5j) = 11

8. Guided Practice Use the dot product to find the length of vector v = 4, –3 (hint: use property 2!!!) Length = 5

9. Guided Practice Find the angle between vectors u and v u = 2, 3 , v = –2, 5 0 = 55.491

10. Guided Practice Find the angle between vectors u and v

11. Guided Practice Find the angle between vectors u and v 0 = 90 Is there an easier way to solve this???

12. Guided Practice Prove that the vectors u = 2, 3 and v = –6, 4 are orthogonal Check the dot product: u v = 0!!! Graphical Support???

13. First, let’s look at a brain exercise… Page 520, #30: Find the interior angles of the triangle with vertices (–4,1), (1,–6), and (5,–1). Start with a graph… A(–4,1) B(5,–1) C(1,–6)

14. First, let’s look at a brain exercise… A(–4,1) B(5,–1) C(1,–6)

15. First, let’s look at a brain exercise… A(–4,1) B(5,–1) C(1,–6)

16. First, let’s look at a brain exercise… A(–4,1) B(5,–1) C(1,–6)

17. Definition: Vector Projection The vector projection of u = PQ onto a nonzero vector v = PS is the vector PR determined by dropping a perpendicular from Q to the line PS. Q u P S v R Thus, u can be broken into components PR and RQ: u = PR + RQ

18. Definition: Vector Projection Q u P S R Notation for PR, the vector projection of u onto v: PR = proj u v The formula: u v proj u = v v 2 |v|

19. Some Practice Problems Find the vector projection of u = 6, 2 onto v = 5, –5 . Then write u as the sum of two orthogonal vectors, one of which is proj u. v Start with a graph…

20. Some Practice Problems… Find the vector projection of u = 6, 2 onto v = 5, –5 . Then write u as the sum of two orthogonal vectors, one of which is proj u. v Start with a graph… u = proj u + u = 2, –2 + 4, 4 v 2

21. Some Practice Problems… Find the vector projection of u = 3, –7 onto v = –2, –6 . Then write u as the sum of two orthogonal vectors, one of which is proj u. v u = proj u + u = –1.8,–5.4 + 4.8,–1.6 v 2

22. Some Practice Problems… Find the vector v that satisfies the given conditions: 2 u = –2,5 , u v = –11, |v| = 10 A system to solve!!!

23. Some Practice Problems… Find the vector v that satisfies the given conditions: 2 u = –2,5 , u v = –11, |v| = 10 OR

24. Some Practice Problems… Now, let’s look at p.520: 34-38 even: What’s the plan??? If u v = 0  orthogonal! If u = kv parallel! 34) Neither 36) Orthogonal 38) Parallel