Download presentation

Presentation is loading. Please wait.

1
Geometry of R2 and R3 Dot and Cross Products

2
Dot Product in R2 Let u = (u1, u2) and v = (v1, v2) then the dot product or scalar product, denoted by u.v, is defined as u.v = u1v1 + u2v2

3
Dot Product in R3 Let u = (u1, u2, u3) and v = (v1, v2, v3) then the dot product or scalar product, denoted by u.v, is defined as u.v = u1v1 + u2v2 + u3v3

4
**Example Find the dot product of each pair of vectors**

u = (-3, 2, -1); v = (-4, -3, 0) u = (-4, 0, -2); v = (-3, -7, 6) u = (-6, 3); v = (5, -8)

5
Theorem 1.2.1 Let u and v be vectors in R2 or R3, and let c be a scalar. Then u.v = v.u c(u.v) = (cu).v = u. (cv) u.(v + w) = u.v + u.w u.0 = 0 u.u = ||u||2 Prove c and e in class.

6
Theorem 1.2.2 Let u and v be vectors in R2 or R3, and let be the angle they form. Then u.v = ||v|| ||u|| cos If u and v are nonzero vectors, then Proof: start with the law of cosines. Convert the sides in terms of the norm. Also expand the norm of v-u and set it equal to the previous.

7
**Example Find the angle between each pair of vectors.**

u = (-1, 2, 3); v = (2, 0, 4) u = (1, 0, 1); v = (-1, -1, 0)

8
Orthogonal Vectors Two vectors u and v in R2 or R3 are orthogonal if u.v = 0. Orthogonal, Normal, and Perpendicular, all mean the same.

9
Theorem 1.2.3 Let u and v be nonzero vectors in R2 or R3 and let be the angle they form. Then is An acute angle if u.v > 0 A right angle if u.v = 0 An obtuse angle if u.v < 0

10
**Cross Product (Only in R3 )**

Let u = (u1, u2, u3) and v = (v1, v2, v3) be nonzero vectors in R3. Then the cross product, denoted by u x v, is the vector (u2v3 – u3v2, u3v1 – u1v3, u1v2 – u2v1)

11
**Cross Product (Convenient notation )**

Let u = (u1, u2, u3) and v = (v1, v2, v3) be nonzero vectors in R3. Then u x v, is the vector obtained by evaluating the determinant:

12
**Example Find the cross product of the following vectors**

u = (-1, 1, 0); v = (2, 3, -1)

13
Theorem 1.2.4 The vector uxv is orthogonal to both u and v.

14
Theorem 1.2.4 Let u, v, and w be vectors in R3, and let c be a scalar. Then u x v = –(v x u) u x (v + w) = (u x v) + (u x w) (u + v) x w = (u x w) + (v x w) c(u x v ) = (cu) x v = u x (cv) u x 0 = 0 x u = 0 u x u = 0 ||u x v|| = ||u|| ||v|| sin = (||u|| ||v|| – ||u.v||2) Do parts c and g.

15
Cross Product: Area Let u, and v, be vectors in R3, Then the area of the parallelogram determined by u and v is ||u x v|| = ||u|| ||v|| sin

16
Example Find the area of the parallelogram determined by the vectors u = (-1, 1, 0) and v = (2, 3, -1).

17
Homework 1.2

Similar presentations

OK

6.4 Vectors and Dot Products Objectives: Students will find the dot product of two vectors and use properties of the dot product. Students will find angles.

6.4 Vectors and Dot Products Objectives: Students will find the dot product of two vectors and use properties of the dot product. Students will find angles.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on human nutrition and digestion article Ppt on osmosis and diffusion Ppt on power grid corporation of india limited Ppt on determinants of economic development Ppt on earth movements and major landforms in italy Ppt on history of badminton in china Ppt on case study analysis Ppt on first conditional and second Ppt on earth hour 2017 Ppt on aircraft emergencies with atc