1. Discrete Math Section 12.4 Define and apply the dot product of vectors Consider the vector equations; (x,y) = (1,4) + t its slope is 3/2 (x,y) = (-2,5) + t its slope is -2/3 Note: 3/2 x -2/3 = -1, so the two vectors are perpendicular. In general, if y 1. y 2 = -1 or y 1. y 2 = - x 1 ∙ x 2 or x 1 ∙x 2 + y 1. y 2 = 0 x 1 x 2

2. Definition of the dot product of two vectors If V = and W = then the dot product of V and W is denoted by V∙W = x 1 ∙x 2 + y 1. y 2 The dot product is also called the scalar product because the dot product of two vectors results in a scalar not another vector. Example: Find v∙w if v = and w =

3. Vector properties 1. Two vectors are perpendicular if their dot product is zero. 2.Two vectors are parallel if v = kw where k is a scalar. 3. The zero vector is parallel and perpendicular to all vectors.

4. example If u =, v =, and w =, find which vectors are parallel and which are perpendicular.

5. Dot product properties 1. u∙ v = v∙ u commutative 2. k(u∙ v ) = (ku)∙ v 3. u∙ (v + w) = u∙ v + u∙ w distributive 4. u∙ u = |u | 2

6. Angle Theorem The angle Θ between two vectors u and v is found by: cos Θ = u ∙ v |u||v| 0 < Θ < 180

7. example Find the angle between the vectors V = and w =

8. assignment Page 444 Problems 2,4,5,8,10,12,14,16