1. Dot Product TS: Developing a capacity for working within ambiguity Warm Up: Copy the below into your notebook 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

3. Given u =, v = and w = i – 4j find each of the below values 2) (w ∙ v)u

4. Given u =, v = and w = i – 4j find each of the below values 3) 4u ∙ v

5. If θ is the angle between two nonzero vectors u and v, then

6. 1)u = 4i & v = -3i Find the angle between the two vectors

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

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: