1. 12.4 Parallel and Perpendicular Vectors: Dot Product

2.  EX. If v = (3, 2), slope= 2/3 V= (x, y)  slope = y/x

3. Parallel Vectors  when slopes can be reduces to be equal. Ex. (4, 2) and (2, 1) Perpendicular Vectors  when product of slope is -1

4. THE DOT PRODUCT v 1 v 2 = x 1 x 2 +y 1 y 2 ** If the result is 0, then the vectors are perpendicular.

5.  Perpendicular vectors are called orthogonal vectors.  If vectors are collinear, they are still considered parallel.  Zero vector is parallel and perpendicular to all vectors.

6. Properties of the Dot Product  uv=vu  uu=|u| 2  k(u  v)= (ku)  v  u (v+w)=u  v + u  w

7. The Angle between 2 Vectors

8. Example Let u = (8, -4) and v = (2, 1) a) Show that u v=v u b) Find the angle between u and v to the nearest tenth.

9. c) Find a vector that is parallel to u. d) Find a vector that is perpendicular to v.

10. Example Given P(0,3), Q(2, 4), R (3, 7), verify that