ch03:vectors - an-najah national university · 1 ch03:vectors vector and scalar quantities adding...
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Ch03:VectorsVector and Scalar quantitiesAdding vectors geometrically and vectors propertiesComponents of vectorsUnit VectorsAdding vectors by componentsMultiplying vectors (scalar and vectoproduct)
3.1: What is physics?In physics we usually deals with many quantities that have both size and direction (vector quantities) like displacement, velocity, acceleration, etc….Not like in chapter two, vector quantities usually exist in more than one dimension.This need a special mathematical language (language of vectors) used by scientists and engineersIn this chapter we will focus on basic language of vectors
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3.2: Scalars and VectorsScalars: quantities only have a single value (only magnitude) and an appropriate unit.
Distance (5 m), time (4 sec.), mass (5 gr.), temperature (15°C),……
Vectors: quantities that have both magnitude and direction coupled with a unit.
Displacement (5 ft. to left), velocity (10 mph, due north), acceleration (9.8 m/s2, downward),….;
A
B
The curved line: distance; the path takenThe red arrow: Displacement vector
Vectors are represented by arrows.
The head of the arrow signifies direction; Tip points away from the starting point in the direction of the vector to the ending point.
Denoted by or
the length of the arrow signifies the magnitude
and is denoted by or
Vectors are equal if they have the same magnitude and direction.
ar a
ar a
3.2: Scalars and Vectors
ar
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Draw vector
Draw vector starting from the tip of
Draw the resultant vector
from the beginning of to the tip of
3.3: Adding vectors geometricallyar
basrrr
+=ar b
r
br
ar
Same is followed for more than two vectors
Cumulative law ( تبديلي):
Associative law (تجميعي):
3.3: Adding vectors geometrically: adding rules
for any order of adding the vectors same result.
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has the same magnitude as but is in the opposite direction.
To subtract , just add the negative of
3.3: Adding vectors geometrically: negative of a vector and vector subtraction
br
− br
0)( =−+ bbrr
ab rrfrom ab rr
to
In solving equations, we must change the sign of the vector when moving it from one side to other
3.3: Adding vectors geometrically: multiplying a vector by a scalar
If the scalar is positive, the direction of the vector does not change but its magnitude is multiplied by the scalar value.If it is negative, its direction is reversed and its magnitude multiplied.
ar2ar ar2−
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3.3: Adding vectors geometrically: Example: adding vectors in drawing
You have the vectors shown below, find the magnitude of the vector sum
to make the sum shown by drawing, we need to use a ruler and a Protractor
Using ruler to measure the magnitude of vector result (note given scale)
Components of a vector projections along coordinate axis (x, y, z)3.4: Components of vectors
a
yaaa
xaa
y
x
r
r
r
r
ofcomponent -y
axisalongvectorofprojection: ofcomponent -x
axisalongvectorofprojection:
⇒
−⇒
−
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From the right angle triangle, we can find:
3.4: Components of vectors
x
y
x
y
aaaaaa
=
=
=
θ
θ
θ
tan
cos
sin
Where θ is the angle between the vector and the +ve x-axis
A plane flies 215 km 22° east of north. How far due east and due north the plane flies?
3.4: Components of vectors: Example
Sol’n: let the displacement be vector
θ from x-axis (east) is 90-22=68°
due east
due north
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3.5: Unit Vectors
Unit vector isA dimensionless vector with a magnitude of 1.
Used only to specify direction.
The represent unit vectors in the x, y and z directions respectively.
1ˆˆˆ === kjikji ˆ,ˆ,ˆ
To identify the direction for vector components, we canuse what is called Unit vector.
3.5: Unit Vectors
²²
ˆˆˆ
yx
yx
AAjAiA
AAA
+
+== r
r
Unit vector can be introduced in any direction like unit vector  in direction of A , where
vector then can berepresented with unit vectors as
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3.6: Adding vectors by components
jaiaa yxˆˆ +=
r jbibb yxˆˆ +=
r
( ) ( )( ) ( )
jrirr
jbaibar
jbibjaiabar
yx
yyxx
yxyx
ˆˆ
ˆˆ
ˆˆˆˆ
+=
+++=
+++=+=
r
r
rrr
( ) ( )xx
yy
x
yyyxxyx ba
barr
θ and babarrr+
+==+++=+= tan)( 2222
Adding vectors is done by adding each component type (x components together and y components together) for the vectors together. Assume the two vectors
yyyxxx barandbar +=+=Hence,
a: Find magnitude and direction ofb: find unit vector in the direction of
if mjibmjia )ˆ4ˆ2( )ˆ2ˆ2( −=+=rr
mjir
jira
)ˆ2ˆ4(
ˆ)42(ˆ)22( )
−=
−++=r
r
°−=−
===>
===>=
==−+=
−
−
2742tan
tantan
5.420²2²4
1
1
θ
θθx
y
x
y
RR
RR
mr
jijibbbb ˆ
204ˆ
202
²4²2
ˆ4ˆ2ˆ ) −+=
−+−
==r
barrrr
+=br
3.6: Adding vectors by components: Example
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3.6: Adding vectors by components: Example
Given the three vectors below, find the resultant cbar rrrr++=
jrirr yxˆˆ +=
r
Rearrange vectors as below
Airplane flies from the originto city A, located 175 km in a direction 30.0° north of east.Next, it flies 153 km 20.0° westof north to city B. Finally, itflies 195 km due west to city C. Find the location of city Crelative to the origin.
3.6: Adding vectors by components: Example
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ExampleSolution: we have 3 vectors as shown; Theresultant
°=−
=
=+−=+=
+−=
+++++=+=
−
8.1123.95
232tan
251)²232()²3.95(²²
ˆ232ˆ3.95
ˆ)(ˆ)(
1
θ
kmRRR
jiR
jcbaicbajRiRR
yx
yyyxxxyxr
r
jijcicjcicc
jijbibjbibb
jijaiajaiaa
yx
yx
yx
ˆ0ˆ195ˆ180sinˆ180cosˆˆ
ˆ144ˆ3.52ˆ110sinˆ110cosˆˆ
ˆ88ˆ152ˆ30sinˆ30cosˆˆ
+−=°+°=+=
+−=°+°=+=
+=°+°=+=
r
r
r
candba rrr ,,cbaR rrrr
++=
3.7: vectors and laws of physicsVectors do not depend on the location of the origin or on the
orientation of the axes; they are all independent of the choice of coordinate system no change in laws of physics
If coordinate system is rotated by an angle φ components changes to without change in the vector. '' and yx aa
and
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φcosabba =⋅rr
Dot product: is the product of a and the projection of vector on vector
3.8 Multiplying vectors: The Scalar or Dot Product of Two Vectors:
φcosbabarrrr
=⋅
abba =⋅rr
br
0. =barr
abba −=rr.
(scalar quantity)
ar
If =0°
If =90°
If =180°
For unit vectors
Dot product characteristics
1) Commutative
2) Distributive in multiplication
0ˆˆˆˆˆˆ =⋅=⋅=⋅ kjkiji
abba rrrr .=⋅( ) cabacba rrrrrrr
⋅+⋅=+⋅
kajaiaa zyxˆˆˆ ++=
r
kbjbibb zyxˆˆˆ ++=
rzzyyxx babababa ++=
rr.
2. aaaaaaaaa zzyyxx =++=rr
1ˆˆˆˆˆˆ =⋅=⋅=⋅ kkjjii ( ⁄⁄ )
(┴)For vectors represented by components
3.8 The Scalar or Dot Product of Two Vectors: properties of scalar product
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Example
ba
barr
rr
and between angle b)
a)
φ
⋅
abbaabba
bababa yyxxrrrr
rr
⋅=⇒=⋅
=+−=+=⋅
−1coscos b)
462 a)
:solution
φφ5²2²1
13²3²2
=+−==
=+==
bb
aar
r
findj.i.b and j.i.aIf ˆ02ˆ01ˆ03ˆ02 +−=+=rr
°==⇒ − 2.60654cos 1φ
bacrrr
×=
φsinabc =
For two vectors and , the vector product is a third vector so that
(if θ = 0° c = 0, if θ = 90° c = ab)
br
ar
The magnitude of the vector iscrcr
direction is perpendicular (┴) to both vectors and cr
br
ar
You can use the right-hand rule to find the direction of . We begin with first vector in the cross product and 0<θ<180°
cr
3.8 Multiplying vectors: The vector or cross Product of Two Vectors:
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abba rrrr×−=×
0=×aa rr
( ) cabacba rrrrrrr×+×=+×
( )dtbdab
dtadba
dtd
rrrrrr×+×=×
jkiik
ijkkj
kijji
ˆˆˆˆˆ
ˆˆˆˆˆ
ˆˆˆˆˆ
=×−=×
=×−=×
=×−=×
k)bab(aj)bab(ai)bab(abbbaaakji
ba xyyxxzzxyzzy
zyx
zyxˆˆˆ
ˆˆˆ
−+−−−==×rr
For vectors represented by components
kajaiaa zyxˆˆˆ ++=
r
kbjbibb zyxˆˆˆ ++=
r
Unit vectors cross product
3.8 The vector or cross Product of Two Vectors: properties
Example
302043
ˆˆˆ
−−=×
kjibarr
kjibaab ˆ8ˆ9ˆ12 ++=×−=×rrrr
?ab ? ba
findkib and jiaIf
=×=×
+−=−=rrrr
rr ˆ3ˆ2ˆ4ˆ3
kji
kjiˆ8ˆ9ˆ12
))2(4)0(3(ˆ))2(0)3(3(ˆ))0(0)3(4(ˆ
−−−=
−−−+−−−−−=
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Example:
axiszandBbetweenangleh
axiszandAbetweenangleg
Aofdirectiontheinvectorunitf
e
BandAbetweenangled
c
b
a
k
−
−
×
⋅
−
+
++−=
+=
r
r
r
rr
rr
rr
rr
rr
r
r
)
)
)
)
)
)
2)
)
find ˆ2ˆ4ˆ and ˆ3ˆ2 If
BA
BA
BA
BA
jiB
jiA
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