02_A Scalars and Vectors

7/17/2019 02_A Scalars and Vectors

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PHYSICS IN LIFE SCIENCES

Scalars and vectors: how todescribe physical quantitiesRelated textbook chapter: 3.5, Appendix C.



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





/ther scalar quantities0

( thermometer (n electric meter

measures temperature

measures energy



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+



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+

http://embryology.med.unsw.edu.au/Defect/images/bodytempgraph.jpg



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+



<=ample o! vector quantity: velocity

2agnitude o! velocity: speed+

3irection o! velocity: elevation angle

separation angle+

Separation angle



<=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'



"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+





<qual Vectors

5he starting point and ending point o! a vector dont matter+



@nequal Vectors

4v5v

54 vv ≠

Same direction but di!!erent magnitude+



@nequal Vectors

1 F 2 F

21 F F ≠

Same magnitude but di!!erent direction+



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+



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



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 !



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



"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



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



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 +=+



!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



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



( )− = + −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 −=⇒=+





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 −=∆



7uestion time

• Iogin to ivle+nus+edu+sg clic& DpollE on the

le!t menu+

• (nswer poll 9*J(J)+



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



!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

+=

+=



"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 +=



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 +



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 +





#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



φ must be less than ).9o+

Night #and Nule %N#N'

• @sing your le!t hand would give the wrongresultG



$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



7uestion time

• Iogin to ivle+nus+edu+sg clic& DpollE on the

le!t menu+

• (nswer poll 9*J(J*+



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+



Orea&

• ;e=t: 3escription o! motion+

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02_A Scalars and Vectors