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Dot Product ES: Developing a capacity for working within ambiguity Warm Up: Look over the below properties The dot product of u = and v =

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Presentation on theme: "Dot Product ES: Developing a capacity for working within ambiguity Warm Up: Look over the below properties The dot product of u = and v =

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 βˆ™ v = u1v1 + u2v2 Properties of the Dot Product Let u, v and w be vectors and let c be a scalar. u βˆ™ v = v βˆ™ u 0 βˆ™ v = 0 u βˆ™ ( v + w) = u βˆ™ v + u βˆ™ w v βˆ™ v = ||v||2 c(u βˆ™ v) = cu βˆ™ v = u βˆ™ cv

2 Given u = <1, 8>, v = <-3, 2> and w = i – 4j find each of the below values
= 𝑒 1 𝑣 𝑒 2 𝑣 2 = 1 βˆ’ =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 βˆ™ v)u Order of Operations! Parenthesis First! = 𝑀 1 𝑣 𝑀 2 𝑣 2 = 1 βˆ’3 + βˆ’4 2 =βˆ’11

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

5 Interested in the proof? It uses Law of Cosines!
If ΞΈ 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 βˆ™ 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 uβˆ™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(Ο€/4)i + sin(Ο€/4)j & v = cos(2Ο€/3)i + sin(2Ο€/3)j Ans: 5Ο€/12


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