1. Dot Product ES: Developing a capacity for working within ambiguity Warm Up: Look over the below properties
The dot product of u = <u1, u2> and v = <v1, v2> is given by u ∙ v = u1v1 + u2v2
Properties of the Dot Product
Let u, v and w be vectors and let c be a scalar.
u ∙ v = v ∙ u
0 ∙ v = 0
u ∙ ( v + w) = u ∙ v + u ∙ w
v ∙ v = ||v||2
c(u ∙ v) = cu ∙ v = u ∙ cv

2. Given u = <1, 8>, v = <-3, 2> and w = i – 4j find each of the below values
= 𝑢 1 𝑣 𝑢 2 𝑣 2
= 1 −
=13
The Dot Product is just a number. By itself it doesn’t really mean anything. However, dot products are used in many other ways.

3. Given u = <1, 8>, v = <-3, 2> and w = i – 4j find each of the below values
2) (w ∙ v)u
Order of Operations! Parenthesis First!
= 𝑤 1 𝑣 𝑤 2 𝑣 2
= 1 −3 + −4 2
=−11

4. Given u = <1, 8>, v = <-3, 2> and w = i – 4j find each of the below values
Order of Operations! Parenthesis First!
= 𝑢 1 𝑣 𝑢 2 𝑣 2
= 1 −
=13

5. Interested in the proof? It uses Law of Cosines!
If θ is the angle between two nonzero vectors u and v, then
Interested in the proof?
It uses Law of Cosines!

6. Find the angle between the two vectors
u = 4i & v = -3i
Draw the picture to check!

7. Find the angle between the two vectors
2) u = <2, -3> & v = <1, -2>
Draw the picture to check!

8. Orthogonal Vectors The vectors u and v are orthogonal if u ∙ v = 0
Note: Orthogonal is the term for
Vectors who have a 90 degree
angle between them.
Makes sense! Given v
And… u

9. Examples:
Are the vectors orthogonal, parallel or neither?
b) u = j, v = i – 2j

10. Examples:
Are the vectors orthogonal, parallel or neither?
u = <15, 3> v = <-5, 25>

11. You Try:
Given u is a whole number vector, where u∙v=11, v = <1,2> and ||u||=5, find u in component form
Ans: <3, 4>
2) Are the following vectors orthogonal, parallel or neither? u = 8i + 4j, v = -2i – j
Ans: parallel
Find the angle between the vectors
u = cos(π/4)i + sin(π/4)j & v = cos(2π/3)i + sin(2π/3)j
Ans: 5π/12