VectorsVectorsChapter 3: Pages 38-57Chapter 3: Pages 38-57

Overview

Common Scalar Common Scalar Quantities:Quantities:•MassMass•WorkWork•EnergyEnergy•PowerPower•TemperatureTemperature•Electric chargeElectric charge

Common Vector Common Vector Quantities:Quantities:•DisplacementDisplacement•VelocityVelocity•AccelerationAcceleration•ForceForce

A vector is a quantity that can be measured in both a magnitude and a direction

Geometric Addition and Subtraction of Vectors

Commutative Law:a + b = b + aAssociative Law:(a + b) + c = a + (b + c)

Vector Subtraction:d = a – b = a + (-b)

Components of a Vector: Resolving the Vectors

ax = a * cos(φ)

ay = a * sin(φ)

a = √(ax2 + ay

2)

tan(φ) = ay/ax

Unit Vector Notation

a = ax i + ay j + az k

b = bx i + by j + bz kAddition by Componentsr = a + brx = ax + bx

ry = ay + by

rz = az + bz

MULTIPLICATION!!!

• Vector by scalar– becomes a new vector of the magnitude

of…– s(scalar) * a– the direction is the original direction of

vector a if s is + and the opposite direction if s is –.

• Vector by vector– Dot product– Cross Product

Dot (scalar) product

• a ∙ b = ab*cos(φ) where φ is the angle between the directions of a and b

• a ∙ b = (ax*bx) + (ay*by) + (az*bz)

Example:

Work = F • d

= (3.0N)(2.0m)ij + (10.5N)(2.0m)ii

= (6J)(0) + (21J)(1)

= 21 J

If an object experiences a constant force, F = (3.0N)i + (10.5N)j in the direction of (2.0m)j, what is the work done by the force?

Cross (Vector) Product• a × b = ab*sin(φ) where φ is the angle

between the directions of a and b• b × a = -(a × b) does not follow a

commutative rule

b × a =

i j k

bx by bz

ax ay az

= (ay•bz - az•by)i + (az•bx - ax•bz)j + (ax•by - ay•bx)k

Knowledge Knowledge QuestionsQuestions

#1-20#1-20

What are the formulas for determining components of a vector?

aaxx = a*cos( = a*cos(θθ))aayy = a*sin( = a*sin(θθ))

a + b = b + a is the _______ law of vector geometric addition.

Commutative LawCommutative Law

When a vector is multiplied by a ______, the direction changes based on that ________ quantity.

Scalar, scalarScalar, scalar

In the formula a•b = ab*cos(φ), φ is __________________.

Angle between the Angle between the directions of a and b.directions of a and b.

True or False:(a) a × b = b × a(b) a • b = b • a

A)A)FalseFalseB)B)TrueTrue

An example of a scalar quantity is _____ and an example of a vector quantity is _____.

A) mass, power C) energy, workB) acceleration, work D) mass, acceleration

D) mass, accelerationD) mass, acceleration

Give the three letters used in expressing direction in unit vector notation.

““I” hat, “j” hat, “k” hatI” hat, “j” hat, “k” hat

Give the formula for vector subtraction.

d = a – b = a + (-b)d = a – b = a + (-b)

A ______ quantity is expressed in both magnitude and direction.

VectorVector

The process of finding the components of a vector is called _______ ___ ______.

Resolving the vectorResolving the vector

Values of ax and ay are given. Find the magnitude of a.

a = (ax2 +ay

2)½

dxx = -25.0 m

dyy = 40.0 mWhat is the magnitude of d?

d = √(dd = √(dxx22 + d + dyy

22) = √(2225 ) = √(2225 mm22) ≈ 47.2 m) ≈ 47.2 m

Torque is given by the formula t = r × F.If a radius is given by (3.0 m)i + (4.0 m)j

and the F = (10N)i + (21N)j, find the torque.

T = r T = r × F = [(3.0m)(21N) – (4.0m)(10N)]k = (23Nm)k

Angular momentum is expressed as l = m(r × v). If an object of mass 200g, and radius (2.0m)i – (2.0m)j is traveling at v = (2.0m/s)i + (3.0m/s)j – (4.0m/s)k, what is its momentum?

l = .200kg*[(3*0 – 4*2)i + (-l = .200kg*[(3*0 – 4*2)i + (-4*2 – 2*0)j + (-2*2 – 3*2)k 4*2 – 2*0)j + (-2*2 – 3*2)k = -1.6 i – 1.6 j – 2.0 k = -1.6 i – 1.6 j – 2.0 k

““Free Response”Free Response”

#1-10#1-10

Name the two laws of vector addition and give their formulas.

Associative LawAssociative Law(a + b) + c = a + (b + c)(a + b) + c = a + (b + c)Commutative LawCommutative Lawa + b = b + aa + b = b + a

Describe two differences between cross and dot products.

1.1.Dot products follow a Dot products follow a commutative rule and cross commutative rule and cross products don’tproducts don’t

2.2.Cross products containing Cross products containing vectors with no k direction vectors with no k direction might result in a vector in the k might result in a vector in the k direction where dot products direction where dot products will not.will not.

Fg = 208 Nθ = 30°Break the force into components.

FFxx = F = Fgg*cos(*cos(θθ) = 180.1 N) = 180.1 NFFyy = F = Fgg*sin(*sin(θθ) = 104 N) = 104 N

θθ

A person walks 3.1 km north, 2.4 km west, and 5.2 km south consecutively. A) what is the magnitude of their displacement? and b) express the displacement vector in unit-vector notation using an appropriate coordinate system.

a)a)√√[(2.4)2 + (2.1)2] = 3.18 km[(2.4)2 + (2.1)2] = 3.18 kmb)(3.1km)i + (2.4km)j – (5.2km)i = (-

2.1km)i + (2.4km)j