medical phyysics-lecture-2 (dr. m fadhali).ppt [compatibility mode]

7/28/2019 Medical Phyysics-Lecture-2 (Dr. M Fadhali).Ppt [Compatibility Mode]

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Vectors...

There are two common ways of indicating thatsomething is a vector quantity:

Boldface notation: A A

Arrow

notation:

AA =

A A

A A

January 10, 2011 Physics 114A - Lecture 5 4/26

The Components of a VectorThe Components of a VectorThe Components of a VectorThe Components of a Vector

Length, angle, and components can becalculated from each other using trigonometry:

cos x A A q = sin y A A q =

2 2 x y A A= +

1tan / x Aq -=


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2D Cartesian and Polar Coordinate Representations

Vector addition

The sum of two vectors is another vector.

A = B + C

B

C A

B

C


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Vector subtraction

Vector subtraction can be defined interms of addition.

B - C

B

C

B

-CB - C

= B + (-1) C

Unit Vectors: A Unit VectorUnit Vector is a vector

having length 1 and no units . It is used to specify a

direction . Unit vector uu points in the

direction of U U . Often denoted with a

hat

: uu =

U U

x

y

z

i i

j j

k k

l Useful examples are the cartesianunit vectors [ i i, j, k , j, k ]

point in the direction of thex , y and z axes.R = r xi + r y j + r zk


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Vector addition using components:

l Consider C C = A A + BB.(a) C C = (A x i i + Ay j j ) + (B x i i + By j j ) = (A x + B x )i i +(Ay + By ) j j

(b) C C = (C x i i + C y j j )

l Comparing components of (a) and (b): C x = A x + B x C y = Ay + By

C C

B x AA

B y B B

Ax

Ay

l Vector A = {0,2,1}l Vector B = {3,0,2}l Vector C = {1,-4,2}

What is the resultant vector, D, fromadding A+B+C?

(a)(a) { {3 3,,- -44,,2 2} } (b)(b) { {44,,- -2 2,,5 5} } (c)(c) { {5 5,,- -2 2,,44} }

Example

D = (AXi + AY j + AZk ) + (B Xi + B Y j + B Zk ) + (C Xi + C Y j + C Zk )

= (AX + BX + C X)i + (AY + BY+ C Y) j + (AZ + B Z + C Z)k

= (0 + 3 + 1) i + (2 + 0 - 4) j + (1 + 2 + 2) k

= {4,-2,5}


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x y z

x y z

A i A j A k

B B i B j B k

= + +

= + +

r

r

x x y y z z

AB

A B A B A B A B

A B Cosq

= + +

=

r r

Dot Product (Scalar Product)

Cross Product (Vector Product)

( )( )( )

( )

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

q

= -

+ -

+ -

=

=

r r

(determinant)

Given two vectors:

Note that , ,

and .

A B A A B B

A B B A

^ ^

r r rr r

r rr r

AB is the magnitude of Btimes the projection of Aon B (or vice versa).

Note that A B = BA

Multiplying Vectors

Describing Position in 3-Space

A vector is used to establish the position of a particle of interest. The position vector, r, locates the particle at somepoint in time.


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January 11, 2011 Physics 114A - Lecture 6 13/2413/24

The Displacement Vector

r xx yy= +r

2 1r r r D = -r r r

2 1

2 2 1 1 ( ) ( )

r r r

x x y y x x y y

xx y y

D = -

= + - +

= D + D

r r r

Instantaneous Velocity in 3D

V = lim ( r / t) as t 0 = dr / dt 3 Components : V x = dx / dt, etc Magnitude, |V| = SQRT( V x2 + Vy 2 + Vz2)

Average Velocity in 3-D

Vavg = (r2 r 1)/(t 2-t1)= r / t

t is scalar so, V vectorparallel to vector


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Properties of VectorsProperties of VectorsProperties of VectorsProperties of Vectors


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We can resolve vector into perpendicular components usingtwo-dimensional coordinate systems:

Polar Coordinates Cartesian Coordinates

cos25.0 (1.50 m)(0.906) 1.36 m xr r = = =

sin25.0 (1.50 m)(0.423) 0.634 m yr r = = =

2 2 2 2 2(1.36 m) (0.634 m) 2.25 m 1.50 m x yr r r = + = + = =

[ ]1 1tan (0.634 m) / (1.36 m) tan (0.466) 25.0q - -= = =

medical phyysics-lecture-2 (dr. m fadhali).ppt [compatibility mode]

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