1. Homework Questions!

2. Section 6.2 Dot Product of Vectors

3. The dot product (inner product) of
U= <U1,U2> and V=<V1,V2> is
U · V = U1V1 + U2V2

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

5. Example 1 Find each dot product A. <3, 4> · <5, 2>
B. <1, -2> · <-4, 3>
C. (2i-j) · (3i-5j)

6. Example 2: Using dot product to find length
Property 2…u · u= |u|2
(Square root both sides)
Use the dot product to find the length of the vector u = <4, -3>
Use the dot product to find the length of the vector u = <5, -12>

7. Angle Between Vectors
cos Ө =
Ө =

8. Example 3: Finding the angle between vectors
Find the angle between the vectors u and v.
u = <2, 3> and v = <-2, 5>
u = <2, 1> and v = <-1, -3>

9. Orthogonal Vectors
The vectors u and v are orthogonal if and only if u · v = 0.
Prove u and v are orthogonal.
u = <2, 3> and v = <-6, 4>

10. Homework
p. 519 (1-24)