Trigonometry The Dot Product LESSON ONE:

Dot Product Mulitply “i”s Multiply “j”s Add products U = <a, b> V = <c, d> U•V = a•c + b•d Xi + Yj = <x, y>

Mulitply “i”s

Multiply “j”s

Add products

Examples U = <7, 5> V = <6, -8> U•V = <7•6 + 5•-8> U•V = <42+ 40> U•V = 82 U = <3, 2> V = <4, -3> U•V = <3•4 + 2•-3> U•V = <12 + -6> U•V = 6 Special Cases “Orthogonal”: Vectors intersect at a right angle; Dot product = 0. U = <5, 0> V = <0, -4> U•V = <5•0 + 0•-4> U•V = <0 + 0> U•V = 0 DP = 0 = 90°

Properties of the Dot Product U•V = V•U (au)•v = a(u•v) = u•(av) (u+v)•w = u•w + v•w |u|² = u•v

U•V = V•U

(au)•v = a(u•v) = u•(av)

(u+v)•w = u•w + v•w

|u|² = u•v

Calculating Components U•V |V| U = <1, 4> V = <-2, 1> 1•-2 + 4•1 √ 4+1 = 2 √ 5 The component of u along v (or the component of u in the direction of v ) is defined as |u|cos ϴ EX: The weight of the car is a vector w that points directly downward. W = U•V

U•V

|V|

U = <1, 4> V = <-2, 1>

1•-2 + 4•1

√ 4+1

= 2

√ 5