MathematicsVectors

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This set of slides may be updated this weekend. I have given you my current slides to get you started on the PreAssignment. Check back here frequently to see if updates have been made. I will post here a record of updates that have been made.

MathematicsVectors

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Definitions

Examples of a Vector and a Scalar

More Definitions

Components, Magnitude and Direction

Unit Vectors and Vector Notation

Vector Math (Addition, Subtraction, Multiplication)

Drawing a Vector

Graphical Vector Math

Symmetry

Sample Problems

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Magnitude: The amount of a quantity represented by a vector or scalar.

Direction: The angle of a vector measured from the positive x-axis going counterclockwise.

Scalar: A physical quantity that has no dependence on direction.

Vector: A physical quantity that depends on direction.

Units: A standard quantity used to determine the magnitude of a vector or value of a scalar.

Here are some helpful definitions.

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N

This is an example of a VectorChange Wind

Speed

Change WindDirection

There are three representations ofa vector.

1. Real life: the actual quantity that the vector represents.

2. Mathematical: a number, with units and a direction.

3. Graphical: an arrow which has a length proportional to the magnitude and a direction the same as the vector.

Real Life

GraphicalRepresentation

MathematicalRepresentation

Magnitude

Direction

Units

24

Northwest

mph

61218

NortheastSoutheastSouthwest

e

s

w

MathematicsVectors

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Change Temperature

GraphicalRepresentation

MathematicalRepresentation

Magnitude

Direction

Units

0

none

degrees C

Degrees C

This is an example of a scalar.

255075100

Real Life

There are three representationsof a scalar as well

1. Real life: the actual quantity that the vector represents.

2. Mathematical: a number, with units and NO direction.

3. Graphical: a point on a graph.

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More Definitions

Component: The projection of a vector along a particular coordinate axis.

Dot Product: The product of two vectors the result of which is a scalar.

Cross Product: The product of two vectors the result of which is another vector.

Right-Hand Rule: The rule which gives the direction of a cross-product.

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cosAAx

sinAAy

x-axis

y-axis

Ax

Ay A

θ

(the vector)

*Note: In order for the equations to work, θ MUST be measured from the positive x-axis going counterclockwise!

To convert from magnitude/direction to components, we use two equations.

(y-component)

(x-component)

(angle*)

(magnitude)

A

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6.43 units 10 units cos50o

7.66 units 10 units sin 50o

x-axis

y-axis

6.43 units

7.66 units 10 units

50o

(the vector)

*Note: In order for the equations to work, θ MUST be measured from the positive x-axis going counterclockwise!

Here is an example.

(y-component)

(x-component)

(angle*)

(magnitude)

A

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22yx AAA

x

y

A

A1tan

To convert from components to magnitude/direction, we use two equations.

x-axis

y-axis

Ax

Ay A

θ

(the vector)

*Note: In order for the equations to work, θ MUST be measured from the positive x-axis going counterclockwise!

(y-component)

(x-component)

(angle*)

(magnitude)

A

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2 210 units = 6.43 units 7.66 units

1 7.66 units50 tan

6.43 unitso

Here is an example.

x-axis

y-axis

6.43 units

7.66 units 10 units

50o

(the vector)

*Note: In order for the equations to work, θ MUST be measured from the positive x-axis going counterclockwise!

(y-component)

(x-component)

(angle*)

(magnitude)

A

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A unit vector is any vector with a magnitude equal to one.

ˆ AA

A

To find a unit vector in same direction as the

vector, divide the vector by its magnitude.

There are three special unit vectors…

1. is a unit vector pointing to the right.2. is a unit vector pointing up.3. is a unit vector pointing forward.

ijk

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Any vector can be written using vector notation.

Vector notation uses the special unit vectors.

ˆˆ ˆx y zA A i A j A k

As an example

ˆˆ ˆ2 units -4 units 0 units

ˆ ˆ 2 units 4 units

A i j k

i j

2 22 units -4 units

4.47 units

A

ˆ ˆ2 units 4 unitsˆ4.47 units

2 units 4 unitsˆ ˆ 4.47 units 4.47 units

ˆ ˆ 0.45 0.89

i jAA

A

i j

i j

2 2ˆ 0.45 0.89 1A

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When adding vectors add their components.

ˆˆ ˆx y zA A i A j A k

ˆˆ ˆ

x y zB B i B j B k

ˆˆ ˆx y zC C i C j C k

ˆˆ ˆ

x x y y z zC A B i A B j A B k

C A B

x x xC A B

y y yC A B

z z zC A B

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Here is an example of adding vectors.

ˆˆ ˆ10 2 3A i j k

ˆˆ2 6B i k

ˆˆ ˆ8 2 9C i j k

ˆˆ ˆ10 2 2 0 3 6C i j k

C A B

8 10 2

2 2 0

9 3 6

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To add vectors when you are given magnitude/direction, convert to components first.

AA A

BB B

ˆ ˆx x y yC A B i A B j

C A B

cosx AA A

siny AA A

cosx BB B

siny BB B

22

x x y yC A B A B

1tany y

Cx x

A B

A B

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Here is an example of adding vectors when only their magnitude and direction are given

20 20oA

10 135oB

ˆ ˆ11.72 13.91C i j

C A B

20cos 20 18.79oxA

20sin 20 6.84oyA

10cos135 7.07oxB

10sin135 7.07oyB

2 211.72 13.91 18.19C

1 13.91tan 49.88

11.72o

C

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There are two ways to multiply vectors, but they cannot be divided

Dot products produce a scalar.

Cross products produce a vector.

A B c

A B C

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When you multiply vectors to get a scalar use a dot product.

ˆˆ ˆx y zA A i A j A k

ˆˆ ˆx y zB B i B j B k

x x y y z zc A B A B A B c A B

If you are given the vectors as components (vector notation)…

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Here is an example of solving a dot product.

10 2 2 0 3 6 2c

ˆˆ ˆ10 2 3A i j k

ˆˆ2 6B i k

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When you multiply vectors to get a scalar use a dot product.

A B cosc AB

AA A

BB B

180 *If then subtract it from 360°

*c A B

If you are given the vectors as magnitude/direction…

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Here is another example of solving a dot product.

020 330 310 180o o o

10 5 cos50 32.14oc

10 20oA

5 330oB

180 *If then subtract it from 360°

*

0360 310 50o o

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When you multiply vectors to get a vector use a cross product. ˆˆ ˆ

x y zA A i A j A k

ˆˆ ˆx y zB B i B j B k

ˆ

ˆ

ˆ

y z z y

z x x z

x y y x

C A B A B i

A B A B j

A B A B k

C A B

x y z z yC A B A B

y z x x zC A B A B

z x y y xC A B A B

If you are given the vectors as components (vector notation)…

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When you multiply vectors to get a vector use a cross product.

ˆˆ ˆ10 2 3A i j k

ˆˆ2 6B i k

ˆ ˆ2 6 3 0 3 2 10 6

ˆ 10 0 2 2

C i j

k

ˆˆ ˆ12 66 4C i j k

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When you multiply vectors to get a vector use a cross product.

AA A

BB B

180 *If then subtract it from 360°

A B sinC AB *C A B

Use the right-hand rule to get the direction.

If you are given the vectors as magnitude/direction…

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When you multiply vectors to get a vector use a cross product.

180 *If then subtract it from 360°

10 5 sin 50 38.3oC

Use the right-hand rule to get the direction.

10 20oA

5 330oB

020 330 310 180o o o *

0360 310 50o o

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1. Point the fingers of your right hand in the direction of the vector A.

2. Curl your fingers toward the direction of the vector B.

3. The cross-product vector C is given by the direction of your thumb.

A

B C

Right-Hand Rule

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A

B

Drawing Vectors

1. Locate the position where the vector is being measured.

2. Draw an arrow, with a tail at the vector position, pointing in the direction of the vector and having a length proportional to its magnitude.

3. Label the vector with its name. Put an arrow above the name or make it boldface.

4. If necessary, move the vector to another position, keeping its length and direction the same.

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A B C

B A C

Graphical Vector Addition

A

CB

B

C

A

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xx AB

yy AB

AB

180B A

x-axis

y-axis

AB

Negative Vectors

B

B A

A

xA

yA

xByB

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A B D

B A D

Graphical Vector Subtraction

AD

B

B

AD

B

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cosA B AB

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

Graphical Dot Product

cosAB

A

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sinA B AB

Graphical Cross Product

sinAA

B

ˆˆ ˆy z z y z x x z x y y xA B A B A B i A B A B j A B A B k

sinA

The magnitude of the cross product is the area of a parallelogram that has the two vectors as its sides.

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If two vectors form a mirror image around one of the axes, then the component of the resultant along that axis is zero.

x-axis

y-axis

A

B

C

Symmetry