Lesson 6-3 The Scalar Product

AP Physics C

6 – 3 The scalar product, or dot product, is a mathematical operation used to determine the component of a given force in the direction of the displacement.

It involves multiplying one vector by a second vector to produce a scalar. This is a combination of two vectors and the cosine of the angle between their directions.

The scalar product can be either A times the component of in the direction of , or B times the component of in the direction of .

A B cosAB

B

A

A

B

Properties of Scalar Products __If __________________Then____________________

Either

Furthermore, Because is parallel to itself Commutative rule of multiplication Distributive rule of multiplication

A and B are

A B = 0

A and B are parallel

A B = AB

A B = 0 A = 0 or B = 0 or A B

2A A = A

A

A B = B A

(A + B) C = A C + B C

The rule for differentiating a dot product (scalar product) is

A B(A B) = B +A

d d d

dt dt dt

A B = x x y y z zA B A B A B

2 2 2 2 Bx y x yA A A B B

1 A Bcos

AB

A B

BA B

p 184 Practice 6-6

Work in scalar-product notation

Definition of work

W = F = F cos = F d dl dl dl

2

1W = F dl

Power – the rate at which a force does work, or the rate of transfer of energy

Power is a scalar quantity like work and energy. The SI unitof power is the watt (W). In the US customary system, the unit of power is the foot-pound per second. A horsepower (hp) is a common multiple of this unit.

A unit of energy, not power, is the kilowatt-hour (kW∙h).

FdW dK

P vdt dt

1 W = 1 J/s

1 hp = 550 ft·lb/s 746 W

61 3.6 10 3.6kW h W s MJ

p 187 Practice 6-8

HW p 198 # 37 – 42, 47, 48 – 50