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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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January 10, 2011 Physics 114A - Lecture 5 5/26
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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January 10, 2011 Physics 114A - Lecture 5 11/26
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 - -= = =