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PHYSICS IN LIFE SCIENCES
Scalars and vectors: how todescribe physical quantitiesRelated textbook chapter: 3.5, Appendix C.
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Key concepts
• What is scalar? What is vector?
• Vector algebra:
– Scalar-vector products – Vector summation tip-to-tail rule
– Vector subtraction
– Scalar product o! vectors
– "ross product o! vectors right-hand rule
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/ther scalar quantities0
( thermometer (n electric meter
measures temperature
measures energy
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Scalar algebra
• Same as the algebra o! numbers+
• Summation: a1b %a b must have the sameunit'+
• Subtraction: a-b %a b must have the sameunit'+
• 2ultiplication: ab %the units o! a and b
combine'• 3ivision: a4b %the units o! a and b combine'
• 5he outcomes are still scalars+
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5he change o! a scalar is still a
scalar
• 6! a scalar quantity is changing with space
or time the di!!erence between two points
is still a scalar+
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7uantity that carries a direction:
position
What is the position o! Shanghai%relative to Singapore'?
•6t is 899 &m away !rom
Singapore+ %magnitude'
•6ts to the ;< o! Singapore%direction'+
7uantities li&e position
have both a magnitude
and a direction+ 5hey are
called vectors+
( vector is represented by an arrow+
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<=ample o! vector quantity: velocity
2agnitude o! velocity: speed+
3irection o! velocity: elevation angle
separation angle+
Separation angle
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<=ample o! vector quantity: !orce
What determines how !ar you can throw the ball?
•#ow hard you push+ %magnitude o! !orce'
•6n what direction you push+ %direction o! !orce'
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"onventions o! writing a vector
x
y
z •6n !igures a vector is representedby an arrow+
v •6n te=t a vector is written as a
letter with an arrow on top or a
bold letter %v '+
• ( vector > its magnitude its direction+
•5he magnitude o! a vector is a scalar + 6t is written as
or simply v %non-bold without the arrow on top'+
v
•5wo vectors are equal only i! they have both the same
magnitude and the same direction+
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<qual Vectors
5he starting point and ending point o! a vector dont matter+
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@nequal Vectors
4v5v
54 vv ≠
Same direction but di!!erent magnitude+
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@nequal Vectors
1 F 2 F
21 F F ≠
Same magnitude but di!!erent direction+
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Vector algebra: Multiplying or
Dividing a Vector by a Scalar
• 5he result is still a vector+
• 5he magnitude o! the vector is multiplied
or divided by the absolute value o! thescalar+
• ( positive scalar does not a!!ect the
direction o! vector+• ( negative scalar reverses the direction o!
the vector+
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Multiply a vector by a scalar A
Aλ>1 A
A
A
A
A
0<λ<1
−1<λ<0
λ=−1
λ<−1
A
A
A
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Multiply a vector by zero?
6t becomes a Aero vector – a point instead o! an
arrow+
• ( Aero vector has no de!ined direction+
•/r it can assume any direction+
ector !
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Vector algebra: new problems
• Summation: A1B = ?
• Subtraction: A-B = ?
• 2ultiplication: AB = ? (only two specialfors will !e intro"#ce"$
• 3ivision: A%B = ? (not to !e intro"#ce"$
• (re the results still vectors?
A B
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"onnect the ending point o!
with the starting point o!
Vector summation
A
B
#ow to de!ine
A
B
B A +
&ectors 'ave a c'aracteristic way of a""in #p)
tip*to*tail r#le
"#$ B A +
A B
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Its only natural
Starting !rom home a man wal&s . &m towards east
then 8 &m towards north+ Whats his position !rom his
home now?
A
B B A +
3oes it ma&e any di!!erence i! he wal&s 8 &m towards
north !irst then . &m towards east?
Birst movement:
Secondmovement
5otal
movement
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Symmetry of vector summation
•Vector summation is commutative+
•Cou are !ree to change the order o! vectors+
A + B
!irst A then B !irst B then A
B + A
A
B
B A + A
B
A B +
A B B A +=+
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!dding more vectors
• (pply Dtip-to-tail ruleE repetitively+• 5he resultant is drawn !rom the starting point
o! the !irst vector to the ending point o! thelast vector+
5hree-vector summation
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More Vectors
• Fust repeat the tip-to-tail
rule+
• 5he resultant is still
drawn !rom the starting
point o! the !irst vector to
the ending point o! the
last vector+
• 5he order o! the vectors
does not matterG
Bour-vector summation
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( )− = + −A B A B
Vector subtraction: two ways
Bind –B and use tip-to-
tail rule+
B
B− B A −
A
B B AC −=
A
Hut the tail o! B and A
together and draw a
vector !rom the tip o! B to tip o! A+
B AC AC B −=⇒=+
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The change of a vector is still a
vector
•6! a vector quantity is changing with space
or time the di!!erence between two points is
still a vector+
1V 2
V
12 V V V −=∆
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7uestion time
• Iogin to ivle+nus+edu+sg clic& DpollE on the
le!t menu+
• (nswer poll 9*J(J)+
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7uestion
• Which o! the !ollowing diagrams is correct?
A B
B A +
A B
B A −−
A B
B A −
A B
B A −
$A# $%#
$C# $&#
! th t d t
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!nother way to draw vector
summation: parallelogram rule
•5he parallelogram rule is equivalent to the tip-
to-tail rule+
•&ector A and B are called DcomponentsE o! C +
A B
B AC
+=
+=
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"rthogonal components
& = , & y are perpendicular to each other hence they arecalled Dorthogonal componentsE o! & +
x
y
xV
yV
222
V V V y x
=+Hythagorean theorem:
y x V V V +=
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Vector summation using components
• Birst !ind or setup an orthogonal coordinate
system+
• <=press the vectors in terms o! orthogonalcomponents+
• (dd corresponding components up result+
• 6t wor&s !or subtraction as wellG
=
y
V*=
V*y
V)y
V)=
V)y1V*y
V)=1V*=
1V
2V
21 V V +
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The product of two vectors
• Scalar product AB also called inner
product or dot product generates a scalar +
• "ross product ALB also called outer
product generates another vector +
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#ross product
θ
.$ 0C0 = M AM MBM sin θ
/$ C is perpendicular to
both A and B.
1$ 2'e "irections of
A, B, C follow t'e
right hand rule.
A
B
B AC ×=
3e!inition o! C > ALB:
3e!ine magnitude
3e!ine direction
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φ must be less than ).9o+
Night #and Nule %N#N'
• @sing your le!t hand would give the wrongresultG
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$roperties of cross product
θ
A 3 B > B 3 A % (nti-symmetricG'
A 3 %B. 1 B/ ' > % A 3 B.' 1 % A 3 B/ ' %associative'
A
B B A×
A, B parallel or anti-parallel A 3 B > 9
We will use it when we learn Dmagnetic !orceE+
θ
B
A B × A
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7uestion time
• Iogin to ivle+nus+edu+sg clic& DpollE on the
le!t menu+
• (nswer poll 9*J(J*+
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7uestion
• 6! vector A points towards west and vectorB points towards south the cross product
ALB points:
%(' towards northwest+%O' towards southwest+
%"' upward+
%3' downward+
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Orea&
• ;e=t: 3escription o! motion+