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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 = is given by u.

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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 = is given by u."— Presentation transcript:

1 Dot Product ES: Developing a capacity for working within ambiguity Warm Up: Look over the below properties The dot product of u = and v = is given by u ∙ v = u 1 v 1 + u 2 v 2 Properties of the Dot Product Let u, v and w be vectors and let c be a scalar. 1. u ∙ v = v ∙ u 2. 0 ∙ v = 0 3. u ∙ ( v + w) = u ∙ v + u ∙ w 4. v ∙ v = ||v|| 2 5. c(u ∙ v) = cu ∙ v = u ∙ cv

2 Given u =, v = and w = i – 4j find each of the below values 1) u ∙ v 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 =, v = and w = i – 4j find each of the below values 2) (w ∙ v)u Order of Operations! Parenthesis First!

4 Given u =, v = and w = i – 4j find each of the below values 3) 4u ∙ v Order of Operations! Parenthesis First!

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

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

7 2) u = & v = Find the angle between the two vectors

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

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

10 Examples: Are the vectors orthogonal, parallel or neither? a)u = v =

11 1)Given u is a whole number vector, where u∙v=11, v = and ||u||=5, find u in component form Ans: 2) Are the following vectors orthogonal, parallel or neither? u = 8i + 4j, v = -2i – j Ans: parallel 3)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 You Try:


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