Physics

Vectors

Physics

Vectors

Javid

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Scalars and VectorsScalars and Vectors

Temperature = ScalarQuantity is specified by a singlenumber giving its magnitude.

Velocity = VectorQuantity is specified by three numbers that give its magnitude and direction(or its components in threeperpendicular directions).

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Properties of VectorsProperties of Vectors

Two vectors are equal if they have the same magnitude and direction.

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Adding VectorsAdding Vectors

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Subtracting VectorsSubtracting Vectors

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Combining VectorsCombining Vectors

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Using the Tip-to-Tail RuleUsing the Tip-to-Tail Rule

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Question to think aboutQuestion to think about

Question: Which vector shows the sum ofA1 + A2 + A3 ?

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Multiplication by a ScalarMultiplication by a Scalar

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Determining the Components of a Vector1. The absolute value |Ax| of the x-component Ax is the

magnitude of the component vector .2. The sign of Ax is positive if points in the positive x-

direction, negative if points in the negative x-direction.

3. The y- and z-components, Ay and Az, are determined similarly.

Coordinate Systemsand Vector Components

Coordinate Systemsand Vector Components

xA xA

xA

Knight’s Terminology:

• The “x-component” Ax is a scalar.

• The “component vector” is avector that always points along the x axis.

• The “vector” is , and it canpoint in any direction.

xA

A

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Determining ComponentsDetermining Components

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Cartesian and PolarCoordinate Representations

Cartesian and PolarCoordinate Representations

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Unit VectorsUnit Vectorsˆ (1,0,0) unit vector in +x-direction = "i-hat"

ˆ (0,1,0) unit vector in +x-direction = "j-hat"

ˆ (0,0,1) unit vector in +z-direction = "k-hat"

i

j

k

ˆˆ ˆ ( , , )x y k x y z x y zA A A A A i A j A k A A A

ˆˆ ˆ4 2 5 (4, 2,5)B i j k Example:

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A = 100 i m= 100 i mB = (= (200 Cos 45200 Cos 4500 i + 200 Cos 45 i + 200 Cos 4500 j ) m) m = ( = (141 i + 141141 i + 141 j ) m) m

Working with VectorsWorking with Vectors

C = A + B = (100 i m) + (-141 i + 141100 i m) + (-141 i + 141 j ) m) m = (-41 i + 141= (-41 i + 141 j ) m ) m

^

^^ ^

^ ^

^^^

C = [Cx2 + Cy

2]½ = [(-41 m)(-41 m)22 + (141 + (141 m))22]½ = 147 m = Tan-1[Cy/|Cx|] = Tan-1[141/41] = 740

Note: Tan-1 ATan = arc-tangent = the angle whose tangent is …

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Tilted AxesTilted Axes

Cx = C Cos

Cy = C Sin

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Arbitrary DirectionsArbitrary Directions

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Perpendicular to a Surface

Perpendicular to a Surface

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Multiplying Vectors*Multiplying Vectors*ˆ ˆ

ˆˆ ˆ

x y z

x y z

A A i A j A k

B B i B j B k

x x y y z z

AB

A B A B A B A B

A B Cos

Dot Product (Scalar Product) Cross Product (Vector Product)

ˆ

ˆ

ˆ

ˆˆ( )

ˆˆ ˆ

y z z y

z x x z

x y y x

AB

x y z

x y z

A B A B A B i

A B A B j

A B A B k

A B Sin a b

i j k

A A A

B B B

A·B is B times the projectionof A on B, or vice versa.

(determinant)

Given two vectors:

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Spherical Coordinates*Spherical Coordinates*

Ax= R Sin Cos Ay= R Sin Sin Az= R Cos

R = [Ax2 + Ay

2 + Az2]½ = |A|

= Tan-1{[Ax2 + Ay

2]½/Az} = Tan-1[Ay/Ax]

R

x

y

z

A

Ax

Ay

Az

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Cylindrical Coordinates*Cylindrical Coordinates*

Ax= r Cos Ay= r Sin Az= z

r = [Ax2 + Ay

2]½

= Tan-1[Ay/Ax]z = Az r

x

y

z

A

Ax

Ay

Az

z

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SummarySummary

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Summary Summary