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