Today in Precalculus

1. Today in Precalculus • Turn in graded wkst and page 511: 1-8 • Notes: Dot Product Angle between Two Vectors Orthogonal & Parallel Vectors • Homework • Quiz Friday

2. Dot Product The dot product of u= u1,u2 and v=v1,v2 is u•v=u1v1 + u2v2 Examples: Find each dot product: 4,-1•8,3 = 32 + -3 = 29 2,-3•-4,-1 = -8 + 3 = -5 4,2•-3,5 = -12 + 10 = -2

3. Properties of Dot Product Let u,v and w be vectors and let c be a scalar. 1. u•v=v•w 2. u•u=|u|2 3. 0•u=0 4. u•(v+w)=u•v + v•w 5. (cu)•v=u•(cv) = c(u•v)

4. Using Properties of Dot Product Find the length of u= -2,4 using dot product. u•u=|u|2 u•u= 4+16 = 20 So |u|2 = 20 Then |u| =

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

6. Angles Between Two Vectors Find the angle between vectors u and v. u = 3,5 v = -2,1 u•v = -6 + 5 = -1

7. Angles Between Two Vectors Find the angle between vectors u and v. u = -1,-3 v = 2,1 u•v = -2 + -3 = -5

8. Orthogonal Vectors If vectors u and v are perpendicular, then u•v = |u||v|cos90°=0 The vectors u and v are orthogonal, then u•v = 0 For non-zero vectors, orthogonal and perpendicular have the same meaning. Zero vectors have no direction angle, so they are not perpendicular to any vector. They are orthogonal to every vector. Ex: Prove u = 3,2 and v = -8,12 are orthogonal. u•v = -24 + 24 = 0

9. Parallel Vectors If vectors u and v are parallel iff: u =kv for some constant k. Ex: Prove u = 3,2 and v = -6,-4 are parallel. -23,2 = -6,-4

10. Proving Vectors are Neither If vectors u and v are not orthogonal or parallel, then they are neither. Show that vectors u and v are neither: u = 3,2 v = -4,-6 u•v = -12 + -12 = -24 ≠ 0

11. Practice Find the dot product: 5,3•12,4 = 60 + 12 = 72 Use the dot product to find |u| if u = 5, -12 so |u| =13 Find the angle θ between u = -4,-3 and v = -1,5

12. Homework Pg 519: 1-4,9,10,13-16,21,22,33-38 Quiz Friday