Dot product

1. 3.3 VECTORS
Dot Product;

2. The Angle Between Vectors
 Let u and v be two nonzero vectors in 2-space or
3-space, and assume these vectors have been
positioned so their initial points coincided. By the
angle between u and v, we shall mean the
angleθ determined by u and v that satisfies 0 ≤
θ ≤ π.

3. Dot Product
 If u and v are non zero vectors in R2 or R3 , and if θ is
the angle between u and v, then the dot product (also
called the Euclidean inner product) of u and v is denoted
by u · v and is defined as
 u · v = ∥u∥∥v∥ cos θ If u = 0 or v = 0, then we define u ·
v to be 0.
 Θ is acute if u·v >0. • θ is obtuse if u·v<0. • θ=π/2 if
u·v=0.

4. Example of Dot product

5. Component Form of the Dot
Product (1/2)

6. Component Form of the Dot
Product (2/2)
 The formula is also valid if u=0 or v=0.

7.  If u and v are nonzero vectors then
it also can be written as
Finding the Angle Between
Vectors

8. Dot Product

9. Example 3
Finding Dot products from Components

10. Theorem 3.3.2
Properties of the Dot Product
If u, v and w are vectors in 2- or 3-space
and k is a scalar, then:

11. Orthogonal Vectors

12. Example 4
A Vector Perpendicular to a Line
 Show that in 2-space the nonzero vector n=(a,b) is
perpendicular to the line ax+by+cz=0.
 Solution

13. Example 5
Vector Component of u Along a

14. Distance between Point &
Plane or line

15. Example 7
Using the Distance Formula

16. Planes in 3-Space
),,( 0000 zyxP 
0)()()(
0
is,thatn,toorthogonalisvectorthe
000
0
0


zzcyybxxa
PPn
PP

17. Example 1
Finding the Point-Normal
Equation of a Plane

18. Distance between Point &
Plane

19. Distance between Parallel
planes

20. 3.5 Lines and Planes
in 3-Space

21. 0)(
so,r-rThenplane.thetonormalvectora
bec)b,(a,nletand)z,y,(xPpointtheto
originthefromvectorthebe),,(let
0
00
0000
0000




rrn
PP
zyxr
Vector Form of Equation of a
Plane

22. Parametric equations of a
plane

23. Example 1
Vector Equation of a Plane

24. Example 2
Parametric Equations of a Line