Dot Product August 2013

Transcription

1 Dot Product August 2013

2 Dot product. v = v 1, v 2,..., v n, w = w 1, w 2,..., w n The dot product v w is v w = v 1 w 1 + v 2 w v n w n n = v i w i. i=1 Example: 1, 4, 5 2, 8, 0 = = 30.

3 Clicker Question: Compute the Dot Product v = 1, 0, 1 u = 0, 1, 1 A. v u = 1, 1, 2 B. v u = 0, 0, 1 C. v u = 1 D. v u = 4 E. I don t understand the question. receiver channel: 41 session ID: bsumath275

4 Dot product is a scalar. The dot product of two vectors is a scalar.

5 Dot product and magnitude. Example: 1, 4, 5 1, 4, 5 = = 1, 4, 5 2. v v = v 2 v = v v

6 Properties of dot product. u v = v u (λ u) v = λ( u v) = u (λ v) u ( v 1 + v 2 ) = u v 1 + u v 2 ( u 1 + u 2 ) v = u 1 v + u 2 v (More properties listed in textbook.)

7 Properties of dot product. u v = v u (commutative) (λ u) v = λ( u v) = u (λ v) u ( v 1 + v 2 ) = u v 1 + u v 2 ( u 1 + u 2 ) v = u 1 v + u 2 v (More properties listed in textbook.)

8 Properties of dot product. u v = v u (commutative) (λ u) v = λ( u v) = u (λ v) u ( v 1 + v 2 ) = u v 1 + u v 2 (distributive) ( u 1 + u 2 ) v = u 1 v + u 2 v (distributive) (More properties listed in textbook.)

9 Dot product and angles. v = v e v, u = u e u u v = u v e u e v. What is the dot product of unit vectors?

10 Dot product and angles. v = v e v, u = u e u u v = u v e u e v. What is the dot product of unit vectors? e v θ e u Say in R 2, e u = 1, 0 and e v = cos θ, sin θ. Then e u e v = cos θ.

11 Dot product and angles. v = v e v, u = u e u u v = u v e u e v. What is the dot product of unit vectors? e v θ e u Say in R 2, e u = 1, 0 and e v = cos θ, sin θ. Then e u e v = cos θ. Conclusion: The dot product of two unit vectors is the cosine of the angle between them.

12 Dot product and angles. v = v e v, u = u e u u v = u v e u e v. What is the dot product of unit vectors? e v θ e u Say in R 2, e u = 1, 0 and e v = cos θ, sin θ. Then e u e v = cos θ. Conclusion: The dot product of two unit vectors is the cosine of the angle between them. If u, v are any vectors and θ is the angle between them, u v = u v cos θ.

13 Dot product and angles. If u, v are any vectors and θ is the angle between them, We usually choose ( 2 ) 0 θ π. E.g.: cos 1 = 2 u v = u v cos θ. ( ) u v θ = cos 1. u v

14 Dot product and angles. If u, v are any vectors and θ is the angle between them, u v = u v cos θ. ( ) u v θ = cos 1. u v We usually choose ( 2 ) 0 θ π. E.g.: cos 1 = π π, not

15 Example: Find the Angle Between two Roads. Two straight roads meet at an intersection. One road, running from south to north, gains 20 feet in elevation for every 100 vertical feet traveled (it has a slope of 1/5). The other, running from west to east, loses 10 feet in elevation for every 100 horizontal feet traveled. What is the angle between these roads at their intersection?

16 Challenge: Points on a Circle. Suppose P and Q are opposite points on a circle. Let R be any other point on the circle. Show that: RP RQ.

17 Orthogonality. u and v are perpendicular or orthogonal (notation: u v) if and only if u v =

18 Orthogonality. u and v are perpendicular or orthogonal (notation: u v) if and only if u v = 0.

19 Orthogonality. u and v are perpendicular or orthogonal (notation: u v) if and only if u v = 0. The angle between u and v is acute θ < π/2 cos θ > 0 u v > 0. The angle between u and v is obtuse θ > π/2 cos θ < 0 u v < 0.

20 Clicker Question: Orthogonality Which one of the following is not perpendicular to the vector: v = 2 î 7 ĵ? A. u = 7 î + 2 ĵ B. u = 2 î + 7 ĵ C. u = 14 î + 4 ĵ + ˆk D. 0 E. I don t understand the question. receiver channel: 41 session ID: bsumath275

21 Clicker Question: Orthogonality Can you find a number a so that the vectors v = a, 2 and u = 0, 3 are orthogonal? A. Yes. B. No. C. I don t understand the question. receiver channel: 41 session ID: bsumath275

22 Projection and decomposition. Let v be a nonzero vector, u any vector. Write: u = u + u, where: u is parallel to v, u is perpendicular to v.

23 Projection and decomposition. Let v be a nonzero vector, u any vector. Write: u = u + u, where: u is parallel to v, u is perpendicular to v. u e v v

24 Projection and decomposition. Let v be a nonzero vector, u any vector. Write: u = u + u, where: u is parallel to v, u is perpendicular to v. u u e v u v

25 Projection and decomposition. Let v be a nonzero vector, u any vector. Write: u = u + u, where: u is parallel to v, u is perpendicular to v. u By trigonometry, u e v u v u = ( u cos θ)e v = ( u e v )e v u v = v e v, and u = u u.

26 Projection and decomposition. u = u v v e v, u = u u. Example: Let v = 3, 12, 4, u = 5, 1, 3.

27 Projection and decomposition. u = u v v e v, u = u u. Example: Let v = 3, 12, 4, u = 5, 1, 3. v = = 169 = 13,

28 Projection and decomposition. u = u v v e v, u = u u. Example: Let v = 3, 12, 4, u = 5, 1, 3. v = = 169 = 13, u v = = 39,

29 Projection and decomposition. u = u v v e v, u = u u. Example: Let v = 3, 12, 4, u = 5, 1, 3. v = = 169 = 13, u v = = 39, u v v = = 3,

30 Projection and decomposition. u = u v v e v, u = u u. Example: Let v = 3, 12, 4, u = 5, 1, 3. v = = 169 = 13, u v = = 39, u v v = = 3, u = u v v e v = , 12 13, 4 13 = 9 13, 36 13,

31 Projection and decomposition. u = u v v e v, u = u u. Example: Let v = 3, 12, 4, u = 5, 1, 3. v = = 169 = 13, u v = = 39, u v v = = 3, u = u v v e v = , 12 13, 4 13 = 9 13, 36 13, u = u u = 56 13, 23 13,

32 Clicker Question: Compute u (wrt v) Find the projection of u along v if: v = 0, 2, 3 u = 1, 1, 0 A. u = 0, 2, 3 B. u = 0, 2/13, 3/13 C. u = 0, 4/ 13, 6/ 13 D. u = 0, 4/13, 6/13 E. I don t understand the question. receiver channel: 41 session ID: bsumath275

33 Exercise: acceleration At a given instant, a car is traveling with velocity v = 25î 10ĵ and acceleration a = 5î + 2ĵ. Find a and a with respect to v. How are a and a experienced by the driver of the car?