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,

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Presentation transcript:

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

Given u = <1, 8>, v = <-3, 2> and w = i – 4j find each of the below values = 𝑢 1 𝑣 1 + 𝑢 2 𝑣 2 = 1 −3 + 8 2 =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.

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 1 + 𝑤 2 𝑣 2 = 1 −3 + −4 2 =−11

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 1 + 𝑢 2 𝑣 2 = 1 −3 + 8 2 =13

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! http://www.youtube.com/watch?v=adOtjgAeIbE

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

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

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

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

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

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