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440:221 Intro to Engineering Mechanics: Statics

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Lecture 3Vector operations

Spring 2014Based on Textbook Material: Engineering Mechanics Statics, 12th Edition, R.C. Hibbeler, Pearson 2010.

Alberto Cuitino, Heather Emady,

Sara Salahi, Bereket YohannesInstructors

Juan Ren, Jingjin Xie

Assistants

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School of Engineering, Spring 2014

Topics to be reviewed

• What is a Cartesian vector representation?

• What is a Cartesian component?• How do we add vectors using Cartesian components?

• What is a unit vector? What is a vector position?

• How do find a force along a given direction?

• What is the inner (or dot) product?

• What is the inner product useful for?

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Scalar Notation – 2D

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F Rx = F cos

F Ry = F sin

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Cartesian Vector Notation – 2D

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• Resolve vectors into components

using the x and y axis system.

• Each component of the vector is

shown as a magnitude and a

direction.

• The directions are based on the xand y axes.

• We use the “unit vectors” i and j to

designate the x and y axes.

F = F x i + F y j

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• Step 1 is to resolve eachforce into its components.

Adding Vectors using Cartesian Components

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• Step 3 is to find the magnitude

and angle of the resultant

vector.

• Step 2 is to add all the x-

components together, followed byadding all the y components

together. These two totals are

the x and y components of the

resultant vector.

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Adding Vectors using Cartesian Components

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Break the three vectors into components, then add them.

FR = F1 + F2 + F3

= F1x i + F1y j F2x i + F2y j + F3x i F3y j

= (F1x F2x + F3x) i + (F1y + F2y F3y) j

= (FRx

) i + (FRy

) j

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Examples

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Examples

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Cartesian Vectors 3D

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A = Ax Ay Az

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Characteristics of a unit vector :

a) Its magnitude is 1.

b) It is dimensionless (has no units).

c) It points in the same direction as the original

vector ( A).

For a vector A, with a magnitude of A, an

unit vector is defined as

u A = A / A .

The unit vectors in the Cartesian axis systemare i, j, and k. They are unit vectors along the

positive x, y, and z axes respectively.

Cartesian unit vectors

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Consider a box with sides AX, AY, and AZ meters

long.

The vector A can be defined as

A = ( AX i + AY j + AZ k)

The projection of vector A in the x-y plane is A´. The magnitude of A´ is

found by using the same approach as a 2-D vector: A´ = (AX2 + AY

2)1/2 .

The magnitude of the position vector A can now be obtained as

A = ((A´)2 + AZ2) ½ = (AX

2 + AY2 + AZ

2) ½

Cartesian vector representation

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These angles are

measured betweenthe vector and the

positive X, Y and Z

axes, respectively.Their range of values

are from 0° to 180°

The direction or orientation of

vector A is defined by theangles , β, and γ.

Direction of a Cartesian vector

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Using trigonometry, “direction cosines” are found using

These angles are not independent.

They must satisfy the following equation:

cos ² + cos ² + cos ² = 1


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This result can be derived from the

definition of a coordinate direction angles

and the unit vector. Recall, the formula

for finding the unit vector of any position

vector:


or written another way,

u A = cos i + cos j + cos k

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For example, if

A = AX i + AY j + AZ k and

B = BX i + BY j + BZ k , then

A + B = (AX + BX) i + (AY + BY) j + (AZ + BZ) k

or

A – B = (AX - BX) i + (AY - BY) j + (AZ - BZ) k .

Once individual vectors are written in Cartesian form, it is

easy to add or subtract them. The process is essentially

the same as when 2-D vectors are added.

Addition of cartesian vectors

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Examples

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Examples

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Vector Position

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Vector Position

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A position vector is defined as a fixed vector that locates a

point in space relative to another point.

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Vector Position

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The position vector directed from A to B, r AB , is defined as

r AB = {( XB – X A ) i + ( YB – Y A ) j + ( ZB – Z A ) k }

Please note that B is the ending point and A is the starting

point. ALWAYS subtract the “tail” coordinates from the “tip”

coordinates!

Consider two points, A and B, in 3-D space.

Let their coordinates be (X A, Y A, Z A) & (XB, YB, ZB ),

respectively.

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Example

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Direction from A to B

Distance

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Example

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Example

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Force Vector Directed Along a Line

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If a force is directed along a line, then we can represent the force vector

in Cartesian coordinates by using a unit vector and the force’s

magnitude. So we need to:

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Procedure

1) Find the position vector, r AB ,

along two points on that line.

2) Find the unit vector describing

the line’s direction, u AB =

(r AB/r AB).

3) Multiply the unit vector by the

magnitude of the force, F = F

u AB .

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Example

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Find Force at A

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Example

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Example

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Dot Product

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A

B

A ▪ B = A B cos

A▪

B = B▪

A Commutativea ( A ▪ B) = (a A) ▪ B = A ▪ (a B) Multipl ication by a scalar

A ▪ (B + D) = ( A ▪ B) + ( A ▪ D) Distributive Law

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Dot Product

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A

B

A ▪ B = Ax Bx + Ay By + Az Bz

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Dot Product Applications

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A

B = cos-1 ( A ▪ B / A B)

1) Angle between two directions

2) Components parallel and perpendicular to a line

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Example

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Find Projections parallel

and perpendicular to AB

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Example

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Example

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440 221 L

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Take-home message

• Addition using Cartesian components is simple

– Add all components in each direction using scalar addition

– Similar procedure for 2 and 3 dimensional cases

– Quite practical for 3D problems

• The direction from A to B is easily determined by

– r AB = {( XB – X A ) i + ( YB – Y A ) j + ( ZB – Z A ) k }

• The inner (dot) product can be easily computed by

– A ▪ B = Ax Bx + Ay By + Az Bz

• The inner product is useful to find

– Angles between directions

– Projections of forces along directions

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