8.5 The Dot Product Precalculus

Definition of the Dot Product If u= and v= are vectors, then their dot product (u v) is defined by: u v = a 1 a 2 + b 1 b 2 Ex: If u = and v=, then u v= (2)(6)+(5)(1) = 17

Properties of the Dot Product The proofs of the following properties of the dot product follow easily from the definition: u v = v u (au) v = a(u v) = u (av) (u+v) w = u w + v w |u| ² = u u

The Dot Product Theorem If θ is the angle between two nonzero vectors u and v, then: u v= |u| |v| cos θ

Angle Between Two Vectors If θ is the angle between two nonzero vectors u and v, then: cos θ= u v |u| |v|

Orthogonal Vectors Two nonzero vectors u and v are perpendicular if and only if u v = 0 Proof: If u and v are perpendicular, then the angle between them is π/ 2 and so u v = |u| |v| cos π/ 2 = 0 Conversely, if u v = 0, then |u| |v| cos θ= 0 Since u and v are nonzero vectors, we conclude that cos θ = 0, and so θ = π/ 2. Thus, u and v are orthogonal. Parallel Vectors : Two vectors are parallel if the angle between them is 0 or π

The Component of u Along v The component of u along v (or the component of u in the direction of v) is defined to be: |u| cos θ (where θ is the angle between u and v) The component of u along v is the magnitude of the portion of u that points in the direction of v

Calculating Components The component of u along v is u v / |v| Ex: let u = and v=. Find the component of u along v. component of u along v= u v / |v|= (1)(- 2)+(4)(1)/ square root of 4+1 = 2/ square root of 5

Work The work W done by a force F in moving along a vector D is W = FD Ex: a force is given by the vector F = and moves an object from the point (1,3) to the point (5,9). Find the work done. D= = W=F D = = 26