Section 9.2Section 9.2

VectorsVectors

GoalsGoals Introduce Introduce vectorsvectors.. Begin to discuss Begin to discuss operationsoperations with vectors with vectors

and vector and vector componentscomponents.. Give Give propertiesproperties of vectors. of vectors.

VectorsVectors

A vector is a quantity (such as displacement or velocity or force) that has both magnitude and direction.

A vector is often represented by an arrow or a directed line segment. The length of the arrow represents the

magnitude of the vector and the arrow points in the direction of the

vector.

Vectors (cont’d)Vectors (cont’d)

For instance, if a particle moves For instance, if a particle moves along a line segment from point along a line segment from point AA to to point point BB, then the corresponding , then the corresponding displacement vectordisplacement vector vv has has initial initial pointpoint AA (the tail) and (the tail) and terminal point terminal point BB (the tip). (the tip).

We writeWe write Note Note

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.ABv

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.AB CD

Vector AdditionVector Addition

If a particle moves from If a particle moves from AA to to BB, and , and then from then from BB to to CC, the net effect is , the net effect is that the particle moves from that the particle moves from AA to to CC..

We writeWe write The next slide givesThe next slide gives

a general definition ofa general definition ofvector addition:vector addition:

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.AC AB BC

Vector Addition (cont’d)Vector Addition (cont’d)

The figure shows The figure shows why this definition why this definition is sometimes called is sometimes called the the Triangle LawTriangle Law..

Vector Addition (cont’d)Vector Addition (cont’d)

We can instead draw another copy of We can instead draw another copy of vv with the same initial point as with the same initial point as uu..

Completing the parallelogram as on the Completing the parallelogram as on the next slide, we see that next slide, we see that uu + + vv = = vv + + uu..

This gives another way to form the sum:This gives another way to form the sum: If we place If we place uu and and vv so they start at the same so they start at the same

point, then point, then uu + + vv lies along the diagonal of the lies along the diagonal of the parallelogram with parallelogram with uu and and vv as sides. as sides.

This is called the Parallelogram Law:

Scalar MultiplicationScalar Multiplication

The following definition shows how we The following definition shows how we multiply a vector by a real number multiply a vector by a real number cc::

Scalar Multiplication Scalar Multiplication (cont’d)(cont’d)

Note that…Note that… Two nonzero vectors are Two nonzero vectors are parallelparallel if they if they

are scalar multiples of one another.are scalar multiples of one another. In particular, the vector –In particular, the vector –vv = (–1) = (–1)vv

(called the (called the negativenegative of of vv)) has the has the same lengthsame length as as vv but but points in the points in the opposite directionopposite direction..

Vector SubtractionVector Subtraction

By the By the differencedifference of two vectors we mean of two vectors we mean

uu – – vv = = uu + (– + (–vv)) We can construct We can construct uu – – vv … …

by first drawing –by first drawing –vv and then adding it to and then adding it to uu by by the Parallelogram Law, orthe Parallelogram Law, or

by means of the Triangle Law.by means of the Triangle Law.

ExampleExample

If If aa and and bb are the vectors shown on the are the vectors shown on the left , draw left , draw aa – 2 – 2bb..

SolutionSolution We first draw the vector –2 We first draw the vector –2bb pointing in the direction opposite to pointing in the direction opposite to bb and and twice as long.twice as long.

We place its tail at the tip of We place its tail at the tip of aa and then and then use the Triangle Law to draw use the Triangle Law to draw aa + (– 2 + (– 2bb):):

ComponentsComponents

If we place the initial point of a vector If we place the initial point of a vector aa at at the origin of a rectangular coordinate the origin of a rectangular coordinate system, then the terminal point of system, then the terminal point of aa has has coordinates of the formcoordinates of the form ((aa11, , aa22) or) or

((aa11, , aa22, , aa33),),

depending on whether our coordinate depending on whether our coordinate system is two- or three-dimensional.system is two- or three-dimensional.

RepresentationsRepresentations

The vectors shown are all equivalent to The vectors shown are all equivalent to the vector terminal the vector terminal point is point is PP(3, 2).(3, 2).

We can think of all these geometric We can think of all these geometric vectors as vectors as representationsrepresentations of the vector of the vector

The particular representationThe particular representationorigin to the point origin to the point PP(3, 2) is called the (3, 2) is called the position vectorposition vector of the point of the point PP : :

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3,2 whoseOP

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from theOP

Representations (cont’d)Representations (cont’d)

In three dimensions,In three dimensions,is the is the position vectorposition vector of the point of the point PP((aa11, , aa22, ,

aa33),.),.

Vector addition leads to the following Vector addition leads to the following result: result:

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1 2 3, ,OP a a aa

MagnitudeMagnitude

The The magnitudemagnitude or or lengthlength of the of the vector vector vv is the length of any of its is the length of any of its representations and is denoted by representations and is denoted by the symbol the symbol ||vv||..

The distance formula gives:The distance formula gives:

Using ComponentsUsing Components

The next slide illustrates the The next slide illustrates the following rules:following rules:

Using Components Using Components (cont’d)(cont’d)

ExampleExample

IfIfthe vectors the vectors aa + + bb, , aa – – bb, 3, 3bb, and 2, and 2aa + 5+ 5bb..

SolutionSolution

4,0,3 and 2,1,5 , find anda b a

4,0,3 and 2,1,5 , find anda b a

The Set The Set VVnn

We denote by…We denote by… VV22 the set of all two-dimensional vectors the set of all two-dimensional vectors

andand VV33 the set of all three-dimensional vectors. the set of all three-dimensional vectors.

We will later need to consider the set We will later need to consider the set VVnn of all of all nn-dimensional vectors.-dimensional vectors.

An An nn-dimensional vector is an ordered -dimensional vector is an ordered nn-tuple-tuple

1 2, , , .na a aa

Properties of VectorsProperties of Vectors

These properties can be verified These properties can be verified either geometrically or algebraically.either geometrically or algebraically.

Standard Basis VectorsStandard Basis Vectors

Three vectors in Three vectors in VV33 play a special role: play a special role:

These vectors have length 1 and point These vectors have length 1 and point in the directions of the positive in the directions of the positive xx-, -, yy-, -, and and zz-axes, as shown on the next slide.-axes, as shown on the next slide. In two dimensions we put In two dimensions we put 1,0 and j 0,1 :i

Standard Basis Vectors Standard Basis Vectors (cont’d)(cont’d)

Standard Basis Vectors Standard Basis Vectors (cont’d)(cont’d)

IfIf

Thus, any vector in Thus, any vector in VV33 can be can be

expressed in terms of expressed in terms of ii, , jj, and , and kk..

1 2 3, , , then we can writea a aa

ExampleExample

If If aa = = ii + 2 + 2jj – 3 – 3kk and and bb = 4 = 4ii + 7 + 7kk, , express the vector 2express the vector 2aa + 3 + 3bb in terms in terms of of ii, , jj, and , and kk..

SolutionSolution

Unit VectorsUnit Vectors

A A unit vectorunit vector is a vector whose is a vector whose length is 1.length is 1. For instance, For instance, ii, , jj, and , and kk are all unit are all unit

vectors.vectors.

If If aa is not the is not the zero vectorzero vector 00, then the , then the unit vector that has the same unit vector that has the same direction as direction as aa is is

1 au a

a a

ExampleExample

Find the unit vector in the direction ofFind the unit vector in the direction of22ii – – jj – 2 – 2kk..

Solution The given vector has lengthSolution The given vector has length

Thus the unit vector with the same Thus the unit vector with the same direction is ⅓ (2direction is ⅓ (2ii – – jj – 2 – 2kk) = ⅔) = ⅔ii - ⅓ - ⅓jj - - ⅔⅔kk..

2 222 2 2 1 2 9 3i j k

ReviewReview

VectorsVectors Combining vectorsCombining vectors

AdditionAddition Scalar multiplicationScalar multiplication

ComponentsComponents Properties of vectorsProperties of vectors