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 =
= 29
2,-3•-4,-1 =
= -5
4,2•-3,5 =
= -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 = = -1

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