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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 = <u1, u2> and v = <v1, v2> is given by u β v = u1v1 + u2v2 Properties of the Dot Product Let u, v and w be vectors and let c be a scalar. u β v = v β u 0 β v = 0 u β ( v + w) = u β v + u β w v β v = ||v||2 c(u β v) = cu β v = u β cv

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**Given u = <1, 8>, v = <-3, 2> and w = i β 4j find each of the below values**

= π’ 1 π£ π’ 2 π£ 2 = 1 β =13 The Dot Product is just a number. By itself it doesnβt really mean anything. However, dot products are used in many other ways.

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**Given u = <1, 8>, v = <-3, 2> and w = i β 4j find each of the below values**

2) (w β v)u Order of Operations! Parenthesis First! = π€ 1 π£ π€ 2 π£ 2 = 1 β3 + β4 2 =β11

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**Given u = <1, 8>, v = <-3, 2> and w = i β 4j find each of the below values**

Order of Operations! Parenthesis First! = π’ 1 π£ π’ 2 π£ 2 = 1 β =13

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**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!

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**Find the angle between the two vectors**

u = 4i & v = -3i Draw the picture to check!

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**Find the angle between the two vectors**

2) u = <2, -3> & v = <1, -2> Draw the picture to check!

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**Orthogonal Vectors The vectors u and v are orthogonal if u β v = 0**

Note: Orthogonal is the term for Vectors who have a 90 degree angle between them. Makes sense! Given v Andβ¦ u

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

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

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You Try: Given u is a whole number vector, where uβv=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 β 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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